Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus Volume 1
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis. 97. y=1+x2 and y=4x2For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 98. y=x,x=4, and y=0For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 99. y=x+2,y=2x1, and x=0For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 100. y=x3 and y=x3For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 101. x=e2y,x=y2,y=0, and y=In(2)For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis. 102. x=9y2,x=ey,y=0, and y=3Yogurt containers can be shaped like frustums. Rotate the line y=1mx around the y-axis to find the volume between y=a and y=b.Rotate the ellipse (x2/a2)+(y2/b2)=1 around the x-axis to approximate the volume of a football, as seen here.Rotate the ellipse (x2/a2)+(y2/b2)=1 around the y-axis to approximate the volume of a football.A better approximation of the volume of a football is given by the solid that comes from rotating y=sinx around the x-axis from x=0 to x= . What is the volume of this football approximation, as seen here?What is the volume of the Bundt cake that comes from rotating y=sinx around the y-axis from x=0 to x= ?For the following exercises, find the volume of the solid described. 108. The base is the region between y=x and y=x2 . Slices perpendicular to the x-axis are semicircles.For the following exercises, find the volume of the solid described. 109. The base is the region enclosed by the generic ellipse (x2/a2)+(y2/b2)=1 . Slices perpendicular to the x-axis are semicircles.Bore a hole of radius a down the axis of a right cone and through the base of radius b, as seen here.Find the volume common to two spheres of radius r with centers that are 2h apart, as shown here.Find the volume of a spherical cap of height h and radius r where h >r, as seen here.Find the volume of a sphere of radius R with a cap of height h removed from the top, as seen here.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 114. [T] Over the curve of y=3x,x=0, and y=3 rotated around the y - axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 115. [T] Under the curve of y=3x,x=0, and x=3 rotated around the y-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 116.[T] Over the curve of y=3x,x=0, and y=3 rotated around the x-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 117. [T] Under the curve of y=3x,x=0, and x=3 rotated around the x-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 118. [T] Under the curve of y=2x3,x=0, and x=2 rotated around the y-axis.For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand. 119. [T] Under the curve of y=2x3,x=0, and x=2 rotated around the x-axis.For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 120. y=1x2,x=0, and x=1For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 121. y=5x3,x=0, and x=1For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 122. y=1x,x=1, and x=100For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 123. y=1x2,x=0, and x=1For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 124. y=11+x2,x=0, and x=3For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 125. y=sinx2,x=0, and x=For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 126. y=11x2,x=0, and x=12For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 127. y=x,x=0, and x=1For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 128. y=(1+x2)3,x=0, and x=1For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. 129. y=5x32x4,x=0, and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 130. y=1x2,x=0, and x=1For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 131. y=x2,x=0 , and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 132. y=ex,x=0, and x=1For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 133. y=In(x),x=1, and x=eFor the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 134. x=11+y2,y=1 , and y=4For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 135. x=1+y2y,y=0, and y=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 136. x=cosy,y=0, and y=For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 137. x=y34y2,x=1 and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 138. x=yey,x=1, and x=2For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and y=0 around the x-axis. 139. x=cosyey,x=0, and x=For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 140. y=3x,y=0,x=0 , and x=2 rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 141. y=x3,y=0, and y=8 rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 142. y=x2,y=x , rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 143. y=x,x=0, and x=1 rotated around the line x=2 .For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 144. y=14x,x=1, and x=2 rotated around the line x=4 .For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 145. y=x and y=x2 rotated around the y-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 146. y=x and y=x2 rotated around the line x=2.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 147. x=y3,y=1x,x=1, and y=2 rotated around the x-axis.For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 148. x=y2 and y=x rotated around the line y=2 .For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 149. [T] Left of x=sin(y) , right of y=x , around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 150. [T] y=x2 and y=4x rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 151.[T] y=cos(x),y=sin(x),x=14, and x=54 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 152. [T] y=x22x,x=2, and x=4 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 153. [T] y=x22x,x=2, and x=4 rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method yon think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 154. [T] y=3x32,y=x, and x=2 rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method yon think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 155. [T] y=3x32,y=x and x=2 rotated around the y-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 156. [T] x=sin(y2) and x=2y rotated around the x-axis.For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume. 157.[T] x=y2,x=y22y+1, and x=2 rotated around the y-axis.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 158. Use the method of shells to find the volume of a sphere of radius r.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 159. Use the method of shells to find the volume of a cone with radius r and height h.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 160. Use the method of shells to find the volume of an ellipse (x2/a2)+(y2/b2)=1 rotated around the x-axis.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 161. Use the method of shells to find the volume of a cylinder with radius r and height h.For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures. 162. Use the method of shells to find the volume of the donut created when the circle x2+y2=4 is rotated around the line x=4 .Consider the region enclosed by the graphs of y=f(x),y=1+f(x),x=0,y=0, and x=a0 . What is the volume of the solid generated when this region is rotated around the y-axis ? Assume that the function is defined over the interval [0,a] .Consider the function y=f(x) , which decreases from f(0)=b to f(1)=0 . Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with x=0 and y=0 , is rotated around the y-axis. Prove that both methods approximate the same volume. Which method is easier to apply? (Hint: Since f(x) is one- to-one, there exists an inverse f1(y) ).For the following exercises, find the length of the functions over the given interval. 165. y=5x from x=0 to x=2For the following exercises, find the length of the functions over the given interval. 166. y=12x+25 from x=1 to x=4For the following exercises, find the length of the functions over the given interval. 167. x=4y from y=1 to y=1Pick an arbitrary lineal function x=g(y) over any interval of your choice (y1,y2) . Determine the length of the function and then prove die length is correct by using geometry.Find the surface area of the volume generated when the curve y=x revolves around the x-axis from (1,1) to (4,2) , as seen here.Find the surface area of the volume generated when the curve y=x2 revolves around the y-axis from (1,1) to (3,9) .For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 171. y=x3/2 from (0,0) to (1,1)For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 172. y=x2/3 from (1,1) to (8,4)For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 173. y=13(x2+2)3/2 from x=0 to x=1For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 174. y=13(x22)3/2 from x=2 to x=4For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 175. [T] y=ex on x=0 to x=1For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 176. y=x33+14x from x=1 to x=3For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 177. y=x44+18x2 from x=1 to x=2For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 178. y=2x3/23x1/22 from x=1 to x=4For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 179. y=127(9x2+6)3/2 from x=0 to x=2For the following exercises, find the lengths of the functions of x over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 180. [T] y=sinx on x=0 to x=For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 181. y=53x4 from y=0 to y=4For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 182. x=12(ey+ey) from y=1 to y=1For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 183. x=5y3/2 from y=0 to y=1For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 184. [T] x=y2 from y=0 to y=1For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 185. x=y from y=0 to y=1For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 186. x=23(y2+1)3/2 from y=1 to y=3For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 187. [T] x=tany from y=0 to y=34For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 188. [T] x=cos2y from y=2 to y=2For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 189. [T] x=4y from y=0 to y=2For the following exercises, find the lengths of the functions of y over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. 190. [T] x=In(y) on y=1e to y=eFor the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 191. y=x from x=2 to x=6For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 192. y=x3 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 193. y=7x from x=1 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 194. [T] y=1x2 from x=1 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 195. y=4x2 from x=0 to x=2For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 196. y=4x2 from x=1 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 197. y=5x from x=1 to x=5For the following exercises, find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 198. [T] y=tanx from x=4 to x=4For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 199. y=x2 from x=0 to x=2For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 200. y=12x2+12 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 201. y=x+1 from x=0 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 202. [T] y=1x from x=12 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 203. y=x3 from x=1 to x=27For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 204. [T] y=3x4 from x=0 to x=1For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 205. [T] y=1x from x=1 to x=3For the following exercises, find the surface area of the volume generated when the following curves revolve around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 206. [T] y=cosx from x=0 to x=2The base of a lamp is constructed by revolving a quarter circle y=2xx2 around the y-axis from x=1 to x=2 , as seen here. Create an integral for the surface area of this curve and compute it.A light bulb is a sphere with radius 1/2 in. with the bottom sliced off to fit exactly onto a cylinder of radius 1/4 in. and length 1/3 in., as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is 1/4 in. Find die surface area (not including the top or bottom of the cylinder).[T] A lampshade is constructed by rotating y=1/x around the x-axis from y=1 to y=2 , as seen here. Determine how much material you would need to construct this lampshade-that is, the surface area-accurate to four decimal places.[T] An anchor drags behind a boat according to the function y=24ex/224 , where y represents the depth beneath the boat and x is the horizontal distance of the anchor from the back of the boat. If the anchor is 23 ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.[T] You are building a bridge that will span 10 ft. You intend to add decorative rope in the shape of y=5|sin((x)/5)| , where x is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot.For the following exercises, find the exact arc length for the following problems over the given interval. 212. y=In(sinx) from x=/4 to x=(3)/4 . (Hint: Recall trigonometric identities.)Draw graphs of y=x2,y=x6 and y=x10 . For y=xn, as n increases, formulate a prediction on the arc length from (0,0) to (1,1) . Now, compute the lengths of these three functions and determine whether your prediction is correct.Compare the lengths of the parabola x=y and the line x=by from (0,0) to (b2,b) as b increases. What do you notice?Solve for the length of x=y from (0,0) to (1,1) . Show that x=(1/2)y2 from (0,0) to (2,2) is twice as long. Graph both functions and explain why this is so.[T] Which is longer between (1,1) and (2,1/2) : the hyperbola y=1/x or the graph of x+2y=3 ?Explain why the surface area is infinite when y=1/x is rotated round the x-axis for 1x , but the volume is finite.For the following exercises, find the work done. 218. Find the work done when a constant force F=12Ib moves a chair from x=0.9 to x=1.1 ft.For the following exercises, find the work done. 219. How much work is done when a person lifts a 50 lb box of comics onto a truck that is 3 ft off the ground?For the following exercises, find the work done. 220. What is the work done lifting a 20 kg child from the floor to a height of 2 m? (Note that 1 kg equates to 9.8 N)For the following exercises, find the work done. 221. Find the work done when you push a box along the floor 2 m, when you apply a constant force of F=100N .For the following exercises, find the work done. 222. Compute the work done for a force F=12/x2N from x=1 to x=2 m.For the following exercises, find the work done. 223. What is the work done moving a particle from x=0 to x=1 m if the force acting on it is F=3x2N ?For the following exercises, find the mass of the one-dimensional object. 224. A wire that is 2 ft long (starting at x=0 ) and has a density function of (x)=x2+2xIb/ft .For the following exercises, find the mass of the one-dimensional object. 225. A car antenna that is 3 ft long (starting at x=0 ) and has a density function of (x)=3x+2Ib/ftFor the following exercises, find the mass of the one-dimensional object. 226. A metal rod that is 8 in. long (stalling at x=0 ) and has a density function of (x)=e1/2xIb/in .For the following exercises, find the mass of the one-dimensional object. 227. A pencil that is 4 in. long (starting at x=2 ) and has a density function of (x)=5/xoz/in .For the following exercises, find the mass of the one-dimensional object. 228. A ruler that is 12 in. long (starting at x=5 ) and has a density function of (x)=In(x)+(1/2)x2oz/in .For the following exercises, find the mass of the two dimensional object that is centered at the origin. 229. An oversized hockey puck of radius 2 in. with density function (x)=x32x+5For the following exercises, find the mass of the two-dimensional object that is centered at the origin. 230. A frisbee of radius 6 in. with density function (x)=ex .For the following exercises, find the mass of the two dimensional object that is centered at the origin. 231. A plate of radius 10 in. with density function (x)=1+cos(x)For the following exercises, find the mass of the two dimensional object that is centered at the origin. 232. A jar lid of radius 3 in. with density function (x)=In(x+1)For the following exercises, find the mass of the two dimensional object that is centered at the origin. 233. A disk of radius 5 cm with density function (x)=3xFor the following exercises, find the mass of the two dimensional object that is centered at the origin. 234. A 12 -in. spring is stretched to 15 in. by a force of 75 lb. What is the spring constant?For the following exercises, find the mass of the two dimensional object that is centered at the origin. 235. A spring has a natural length of 10 cm. It takes 2 J to stretch the spring to 15 cm. How much work would it take to stretch the spring from 15 cm to 20 cm?For the following exercises, find the mass of the two dimensional object that is centered at the origin. 236. A 1 -m spring requires 10 J to stretch the spring to 1.1 m. How much work would it take to stretch the spring from 1 m to 1.2 m?For the following exercises, find the mass of the two dimensional object that is centered at the origin. 237. A spring requires 5 J to stretch the spring from 8 cm to 12 cm, and an additional 4 J to stretch the spring from 12 cm to 14 cm. What is the natural length of the sp ring?For the following exercises, find the mass of the two dimensional object that is centered at the origin. 238. A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?A force of F=20xx3N stretches a nonlinear spring by x meters. What work is required to stretch the spring from x=0 to x=2 m?Find the work done by winding up a hanging cable of length 100 ft and weight-density 5 lb/ft.For the cable in the preceding work is done to lift the cable 50 ft?For the cable in the preceding exercise, how much additional work is done by hanging a 200 lb weight at the end of the cable?[T] A pyramid of height 500 ft has a square base 800 ft by 800 ft. Find the area A at height h. If the rock used to build the pyramid weighs approximately w=100Ib/ft3 , how much work did it take to lift all the rock?[T] For the pyramid in the preceding exercise, assume there were 1000 workers each working 10 hours a day, 5 days a week, 50 weeks a year. If the workers, on average, lifted 10 100 lb rocks 2ft/hr , how long did it take to build the pyramid?[T] The force of gravity on a mass m is F=((GMm)/x2) newtons. For a rocket of mass m = 1000 kg, compute the work to lift the rocket from x=6400 to x=6500 km. (Note: G=61017Nm2/kg2 and M=61024kg .)[T] For the rocket in the preceding exercise, find the work to lift the rocket from x=6400 to x= .[T] A rectangular dam is 40 ft high and 60 ft wide. Compute the total force F on the dam when the surface of tine water is at the top of the dam and the surface of die water is halfway down the dam.[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius 5 ft and height 200 ft. Use the fact that the density of water is 62 lb/ft3.[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.[T] How much work is required to pump out a swimming pool if the area of the base is 800 ft , the water is 4 ft deep, and the top is 1 ft above the water level? Assume that the density of water is 62 lb/ft3.A cylinder of depth H and cross-sectional area A stands full of water at density . Compute the work to pump all the water to the top.For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.A cone-shaped tank has a cross-sectional area that increases with its depth: A=(r2h2)/H3 . Show that the work to empty it is half the work for a cylinder with the same height and base.Compute the area of each of the three sub-regions. Note that the areas of regions R2 and R3 should include the areas of the legs only, not the open space between them. Round answers to the nearest square foot.Determine the mass associated with each of the three sub-regions.Calculate the center of mass of each of the three sub-regions.Now, treat each of the three sub-regions as a point mass located at the center of mass of the corresponding sub-region. Using this representation, calculate the center of mass of the entire platform.Assume the visitor center weighs 2,200,000 lb, with a center of mass corresponding to the center of mass of R3 . Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change?Although the Skywalk was built to limit the number of people on the observation platform to 120, the platform is capable of supporting up to 800 people weighing 200 lb each. If all 800 people were allowed on the platform, and all of them went to the farthest end of the platform, how would the center of gravity of the system be affected? (Include the visitor center in the calculations and represent the people by a point mass located at the farthest edge of the platform, 70 ft from the canyon wall.)For the following exercises, calculate the center of mass for the collection of masses given. 254. m1=2 at x1=1 and m2=4 at x2=2For the following exercises, calculate the center of mass for the collection of masses given. 255. m1=1 at x1=1 and m2=3 at x2=2For the following exercises, calculate the center of mass foi the collection of masses given. 256. m=3 at x=0,1,2,6For the following exercises, calculate the center of mass for the collection of masses given. 257. Unit masses at (x,y)=(1,0)(0,1)(1,1)For the following exercises, calculate the center of mass for the collection of masses given. 258. m1=1 at (1,0) and m2=4 at (0,1)For the following exercises, calculate the center of mass for the collection of masses given. 259. m1=1 at (1,0) and m2=3 at (2,2)For the following exercises, compute the center of mass x . 260. =1 for x(1,3)For the following exercises, compute the center of mass x . 261. =x2 for x(0,L)For the following exercises, compute the center of mass x . 262. =1 for x(0,1) , and =2 for x(1,2)For the following exercises, compute the center of mass x . 263. =sinx for x(0,)For the following exercises, compute the center of mass x . 264. =cosx for x(0,2)For the following exercises, compute the center of mass x . 265. =ex for x(0,2)For the following exercises, compute the center of mass x . 266. =x3+xex for x(0,1)For the following exercises, compute the center of mass x . 267. =xsinx for x(0,)For the following exercises, compute the center of mass x . 268. =x for x(1,4)For the following exercises, compute the center of mass x . 269. =Inx for x(1,e)For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 270. =7 in the square 0x1,1y1For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 271. =3 in the triangle with vertices (0,0) , (a,0) , and (0,b)For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 272. =2 for the region bounded by y=cos(x) , y=cos(x) , x=2 , and x=2For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 273. [T] The region bounded by y=cos(2x) , x=4 , and x=4For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 274. [T] The region between y=2x2 , y=0 , x=0 , and x=1For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 275. [T] The region between y=54x2 and y=5For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 276. [T] Region between y=x , y=In(x) , x=1 and x=4For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 277. [T] The region bounded by y=0,x24+y29=1For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 278. [T] The region bounded by y=0,x=0 , and x24+y29=1For the following exercises, compute the center of mass (x,y) . Use symmetry to help locate the center of mass whenever possible. 279. [T] The region bounded by y=x2 and y=x4 in the first quadrantFor the following exercises, use the theorem of Pappus to determine the volume of the shape. 280. Rotating y=mx around the x-axis between x=0 and x=1For die following exercises, use the theorem of Pappus to determine the volume of the shape. 281. Rotating y=mx around the y-axis between x=0 and x=1For the following exercises, use the theorem of Pappus to determine the volume of the shape. 282. A general cone created by rotating a triangle with vertices (0,0) , (a,0) , and (0,b) around the y -axis. Does your answer agree with the volume of a cone?A general cylinder created by rotating a rectangle with vertices (0,0),(a,0),(0,b) , and (0,0),(a,0),(0,b) around the y -axis. Does your answer agree with the volume of a cylinder?A sphere created by rotating a semicircle with radius a around the y -axis. Does your answer agree with the volume of a sphere?For the following exercises, use a calculator to draw the legion enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 285. [T] Quarter-circle: y=1x2,y=0 , and x=0For the following exercises, use a calculator to draw the legion enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 286. [T] Triangle: y=x,y=2x , and y=0For the following exercises, use a calculator to draw the legion enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 287. [T] Lens: y=x2 and y=xFor the following exercises, use a calculator to draw the legion enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 288. [T] Ring: y2+x2=1 and y2+x2=4For the following exercises, use a calculator to draw the legion enclosed by the curve. Find the area M and the centroid (x,y) for the given shapes. Use symmetry to help locate the center of mass whenever possible. 289. [T] Half-ring: y2+x2=1,y2+x2=4 , and y=0Find the generalized center of mass in the sliver between y=xa and y=xb with ab. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.Find the generalized center of mass between y=a2x2,x=0, and y=0 . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.Find the generalized center of mass between y=bsin(ax) , x=0 , and x=a . Then, use the Pappus theorem to find the volume of the solid generated when revolving around they-axis.Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x=b, b0 , and is rotated around the y-axis.Find the center of mass (x,y) for a thin wire along the semicircle y=1x2 with unit mass. (Hint: Use the theorem of Pappus.)For the following exercises, find the derivative dydx . 295. y=In(2x)For the following exercises, find the derivative dydx . 296. y=In(2x+1)For the following exercises, find the derivative dydx . 297. y=1InxFor the following exercises, find the indefinite integral. 298. dt3tFor the following exercises, find the indefinite integral. 299. dx1+x dx/ 1 + xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 300. [T] y=In(x)xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 301. [T] y=xIn(x)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 302. [T] y=log10xFor the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 303. [T] y=In(sinx)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 304. [T] y=In(Inx)the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 305. [T] y=7In(4x)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 306. [T] y=In(( 4x)7)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 307. [T] y=In(tanx)For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 308. [T] y=In(tan(3x))For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) 309. [T] y=In(cos2x)For the following exercises, find the definite or indefinite integral. 310. 01dx3+xFor the following exercises, find the definite or indefinite integral. 311. 01dt3+2tFor the following exercises, find the definite or indefinite integral. 312. 02xdx x 2+1For the following exercises, find the definite or indefinite integral. 313. 02 x 3dx x 2+1For the following exercises, find the definite or indefinite integral. 314. 2edxxInxFor the following exercises, find the definite or indefinite integral. 315. 2edx ( xIn( x ) ) 2For the following exercises, find the definite or indefinite integral. 316. cosxdxsinxFor the following exercises, find the definite or indefinite integral. 317. 0/4tanxdxFor the following exercises, find the definite or indefinite integral. 318. cot(3x)dxFor the following exercises, find the definite or indefinite integral. 319. ( Inx ) 2xdxFor the following exercises, compute dy/dx by differentiating Iny . 320. y=x2+1For the following exercises, compute dy/dx by differentiating Iny . 321. y=x2+1x21For the following exercises, compute dy/dx by differentiating Iny . 322. y=esinxFor the following exercises, compute dy/dx by differentiating Iny . 323. y=x1/xFor the following exercises, compute dy/dx by differentiating Iny . 324. y=e(ex)For the following exercises, compute dy/dx by differentiating Iny . 325. y=xeFor the following exercises, compute dy/dx by differentiating Iny . 326. y=x(ex)For the following exercises, compute dy/dx by differentiating Iny . 327. y=xxx63For the following exercises, compute dy/dx by differentiating Iny . 328. y=x1/InxFor the following exercises, compute dy/dx by differentiating Iny . 329. y=eInxFor the following exercises, evaluate by any method 330. 510dtt5x10xdttFor the following exercises, evaluate by any method 331. 1edxx+21dxxFor the following exercises, evaluate by any method 332. ddxx1dttFor the following exercises, evaluate by any method 333. ddxxx2dttFor the following exercises, evaluate by any method. 334. ddxIn(secx+tanx)For the following exercises, use the function Inx . If you are unable to find intersection points analytically, use a calculator. 335. Find the area of the region enclosed by x=1 and y=5 above y=Inx .For the following exercises, use the function Inx . If you are unable to find intersection points analytically, use a calculator. 336.[T] Find the arc length of Inx from x=1 to x=2.For the following exercises, use the function Inx . If you are unable to find intersection points analytically, use a calculator. 337. Find the area between Inx and the x -axis from x=1 to x=2 .Find the volume of the shape created when rotating this curve from x=1 to x=2 around the x-axis, as pictured here.[T] Find the surface area of the shape created when rotating the curve in the previous exercise from . x=1 to x=2 around the x-axis.If you are unable to find intersection points analytically in the following exercises, use a calculator. 340. Find the area of die hyperbolic quarter-circle enclosed by x=2 and y=2 above y=1/x .If you are unable to find intersection points analytically in the following exercises, use a calculator. 341. [T] Find the arc length of y=1/x from x=1 to x=4 .If you are unable to find intersection points analytically in the following exercises, use a calculator. 342. Find the area under y=1/x and above the x-axis from x=1 to x=4 .For the following exercises, verify the derivatives and antiderivatives. 343. ddxIn(x+x2+1)=11+x2For the following exercises, verify the derivatives and antiderivatives. 344. ddxIn(xax+a)=2a(x2a2)For the following exercises, verify the derivatives and antiderivatives. 345. ddxIn(1+ 1 x 2 x)=1x1x2For the following exercises, verify the derivatives and antiderivatives. 346. ddxIn(x+x2a2)=1x2a2For the following exercises, verify the derivatives and antiderivatives. 347. dxxIn( x)In( Inx)=In(In(Inx))+CTrue or False ? If true, prove it. If false, find the true answer 348. The doubling time for y=ect is (In(2)/In(c)) .True or False ? If true, prove it. If false, find the true answer. 349. If you invest $500, an annual rate of interest of 3% yields more money in the first year than a 2.5% continuous rate of interest.True or False ? If true, prove it. If false, find the true answer. 350. If you leave a 100°C pot of tea at room temperature (25°C) and an identical pot in the refrigerator (5°C), with k=0.02 , the tea in the refrigerator reaches a drinkable temperature (70°C) more than 5 minutes before the tea at room temperature.True or False ? If true, prove it. If false, find the true answer. 351. If given a half-life of t years, the constant k for y=ekt is calculated by k=In(1/2)/t .For the following exercises, use y=y0ekt . 352. If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by 10?For the following exercises, use y=y0ekt . 353. If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by 100 ?For the following exercises, use y=y0ekt . 354. How old is a skull that contains one-fifth as much radiocarbon as a modem skull? Note that the half-life of radiocarbon is 5730 years.For the following exercises, use y=y0ekt . 355. If a relic contains 90% as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years.For the following exercises, use y=y0ekt . 356. The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.For the following exercises, use y=y0ekt . 357. The populations of New York and Los Angeles are growing at 1% and 1.4% a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?Suppose the value of 1 in Japanese yen decreases at 2% per year. Starting from 1=250 , when will 1=1 ?The effect of advertising decays exponentially. If 40% of the population remembers a new product after 3 days, how long will 20% remember it?If y=1000 at t=3 and y=3000 at t=4 , what was y0 at t=0 ?If y=100 at t=4 and y=10 at t=8 , when does t=1 ?If a bank offers annual interest of 7.5% or continuous interest of 7.25%, which has a better annual yield?What continuous interest rate has the same yield as an annual rate of 9%'?If you deposit 5000 at 8% annual interest, how many years can you withdraw 500 (starting after die first year) without running out of money?You are trying to save 50,000 in 20 years for college tuition for your child. If interest is a continuous 10%, how much do you need to invest initially?You are cooling a turkey that was taken out of the oven with an internal temperature of 165°F. After 10 minutes of resting the turkey in a 70°F apartment, the temperature has reached 155°F. What is the temperature of the turkey 20 minutes after taking it out of the oven?You are trying to thaw some vegetables that are at a temperature of 1°F. To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of 44°F. You check on your vegetables 2 hours after putting them in the refrigerator to find that they are now 12°F. Plot the resulting temperature curve and use it to determine when the vegetables reach 33°F.You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era (146 million years to 65 million years ago), and you find by radiocarbon dating that there is 0.000001% the amount of radiocarbon. Is this bone from the Cretaceous?The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of 24,000 years. If 1 barrel containing 10 kg of plutonium-239 is sealed, how many years must pass until only 10g of plutonium-239 is left?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 370.[T] The best-fit exponential curve to the data of the form P(t)=aebt is given by P(t)=2686e0.01604t . Use a graphing calculator to graph the data and the exponential curve together.For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 371. [T] Find and graph the derivative y of your equation. Where is it increasing and what is the meaning of this increase?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 372. [T] Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?For the next set of exercises, use the following table, which features the world population by decade. Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849 Source: http://www.factmonster.com/ipka/A0762181.html. 373. [T] Find the predicted date when the population reaches 10 billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since1850 Population(thousands) 0 21.00 10 56.80 20 149.5 30 234.0 Source: http://www.sfgenealogy.com/sf/history/hgpop.htm. 374. [T] The best-fit exponential curve to the data of the form P(t)=aebt is given by P(t)=35.26e0.06407t , Use a graphing calculator to graph the data and the exponential curve together.For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since1850 Population(thousands) 0 21.00 10 56.80 20 149.5 30 234.0 Source: http://www.sfgenealogy.com/sf/history/hgpop.htm. 375. [T] Find and graph the derivative y of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Years since1850 Population(thousands) 0 21.00 10 56.80 20 149.5 30 234.0 Source: http://www.sfgenealogy.com/sf/history/hgpop.htm. 376. [T] Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?[T] Find expressions for coshx+sinhx and coshxsinhx . Use a calculator to graph these functions and ensure your expression is correct.From the definitions of cosh(x) and sinh(x) , find their antiderivatives.Show that cosh(x) and sinh(x) satisfy y=y .Use the quotient rule to verify that tanh(x)=sech2(x) .Derive cosh2(x)+sinh2(x)=cosh(2x) from the definition.Take the derivative of the previous expression to find an expression for sinh(2x) .Prove sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y) by changing the expression to exponentials.Take the derivative of the previous expression to find an expression for cosh(x+y) .For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 385. [T] cosh(3x+1)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 386. [T] sinh(x2)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 387. [T] 1cosh(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 388. [T] sinh(In(x))For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 389. [T] cosh2(x)+sinh2(x)For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct. 390. [T] cosh2(x)sinh2(x)