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All Textbook Solutions for Calculus: Early Transcendentals

Refer to the figure and find the volume generated by rotating the given region about the specified line. R1 about OC Refer to the figure and find the volume generated by rotating the given region about the specified line. R1 about AB Refer to the figure and find the volume generated by rotating the given region about the specified line. R1 about BC Refer to the figure and find the volume generated by rotating the given region about the specified line. R2 about OA Refer to the figure and find the volume generated by rotating the given region about the specified line. R2 about OC Refer to the figure and find the volume generated by rotating the given region about the specified line. R2 about AB Refer to the figure and find the volume generated by rotating the given region about the specified line. R2 about BC Refer to the figure and find the volume generated by rotating the given region about the specified line. R3 about OA Refer to the figure and find the volume generated by rotating the given region about the specified line. R3 about OC Refer to the figure and find the volume generated by rotating the given region about the specified line. R3, about AB Refer to the figure and find the volume generated by rotating the given region about the specified line. R3, about BC Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. y=ex2, y = 0, x = 1, x = 1 (a) About the x-axis (b) About y = 1 Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. y = 0, y = cos2x, / 2 x / 2 (a) About the x-axis (b) About y = 1 Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. x2 + 4y2 = 4 (a) About y = 2 (b) About x = 2 Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. y = x2, x2 + y2 = 1, y 0 (a) About the x-axis (b) About the y-axis Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves. y = ln(x6 + 2), y=3x3 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves. y=1+xex3, y = arctan x2 Each integral represents the volume of a solid. Describe the solid. 0sinxdx Each integral represents the volume of a solid. Describe the solid. 11(1y2)2dy Each integral represents the volume of a solid. Describe the solid. 01(y4y8)dy Each integral represents the volume of a solid. Describe the solid. 14[32(3x)2]dx A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94,106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.A log 10 m long is cut at 1-meter intervals and its cross-sectional areas A (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with n = 5 to estimate the volume of the log.(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with n = 4 to estimate the volume of the solid. (b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with n = 4.Find the volume of the described solid S. A right circular cone with height h and base radius r Find the volume of the described solid S. A frustum of a right circular cone with height h, lower base radius R, and top radius r Find the volume of the described solid S. A cap of a sphere with radius r and height h Find the volume of the described solid S. A frustum of a pyramid with square base of side b, square top of side a, and height h What happens if a = b? What happens if a = 0?Find the volume of the described solid S. A pyramid with height h and rectangular base with dimensions b and 2b Find the volume of the described solid S. A pyramid with height h and base an equilateral triangle with side a (a tetrahedron)Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm Find the volume of the described solid S. The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares.Find the volume of the described solid S. The base of S is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.Find the volume of the described solid S. The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-sections perpendicular to the y-axis are equilateral triangles.Find the volume of the described solid S. The base of S is the same base as in Exercise 56, but cross-sections perpendicular to the x-axis are squares.Find the volume of the described solid S. The base of S is the region enclosed by the parabola y = 1 x2 and the x-axis. Cross-sections perpendicular to the y-axis are squares.Find the volume of the described solid S. The base of S is the same base as in Exercise 58, but cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base.Find the volume of the described solid S. The base of S is the region enclosed by y = 2 x2 and the x-axis. Cross-sections perpendicular to the y-axis are quarter-circles.Find the volume of the described solid S. The solid S is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola y=12(1x2),1x1 The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are isosceles triangles with height h and unequal side in the base. (a) Set up an integral for the volume of S. (b) By interpreting the integral as an area, find the volume of S.(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R. (b) By interpreting the integral as an area, find the volume of the torus.64E (a) Cavalieris Principle states that if a family of parallel planes gives equal cross-sectional areas for two solids S1, and S2, then the volumes of S1, and S2 are equal. Prove this principle. (b) Use Cavalieris Principle to find the volume of the oblique cylinder shown in the figure.Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.67E A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.A hole of radius r is bored through the middle of a cylinder of radius R r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.A hole of radius r is bored through the center of a sphere of radius R r. Find the volume of the remaining portion of the sphere.Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas. (a) A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R cx2, h/2 x h/2, where c is a positive constant. Show that the radius of each end of the barrel is r = R d, where d = ch2/ 4. (b) Show that the volume enclosed by the barrel is V=13h(2R2+r225d2)Suppose that a region has area A and lies above the x-axis. When . is rotated about the x-axis, it sweeps out a solid with volume V1, When is rotated about the line y = k (where k is a positive number), it sweeps out a solid with volume V2. Express V2 in terms of V1, k, and A.Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What are its circumference and height? Use shells to find V.Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Explain.Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y=x,3 y = 0, x = 1 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = x3, y = 0, x = 1, x = 2 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y=ex2, y = 0, x = 0, x =1 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = 4x x2, y = x Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = x2, y = 6x 2x2 Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y=x and y = x2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 1, x = 0, y = 1, y = 3 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y=x, x = 0, y = 2 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y=x3/2, y = 8, x = 0 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 3y2 + 12y 9, x = 0 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 1 + (y 2)2, x = 2 Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. x + y = 4, x = y2 4y + 4 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y = x3, y = 8, x = 0; about x = 3 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y = 4 2x, y = 0, x = 0; about x = 1 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y = 4x x2, y = 3; about x = 1 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. y=x, x = 2y; about x = 5 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. x = 2y2, y 0, x = 2; about y = 2 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. x = 2y2, x = y2 + 1; about y = 2 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. y = xex, y = 0, x = 2; about the y-axis (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. y = tan x, y = 0, x = /4; about x = /2 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. y = cos4x, y = cos4x, /2 x /2; about x =(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. y = x, y = 2x/(l + x3); about x = 1 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. x=siny, 0 y , x = 0; about y = 4 (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. x2 y2 = 7, x = 4; about y = 5 Use the Midpoint Rule with n = 5 to estimate the volume obtained by rotating about the y-axis the region under the curve y=1+x3, 0 x 1.If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with n = 5 to estimate the volume of the solid.29E 30E 31E 32E Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves. y = x2 2x, y=xx2+1 34E The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y = x2 + 6x 8, y = 0; about the y-axis The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y = x2 + 6x 8, y = 0; about the x-axis The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y2 x2 = 1, y = 2; about the x-axis The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. y2 x2 = 1, y = 2; about the y-axis The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x2 + (y 1)2 = 1; about the y-axis The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x = (y 3)2, x = 4; about y = 1 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. x = (y 1)2, x y = 1; about x = 1 Let T be the triangular region with vertices (0, 0), (1, 0), and (1, 2), and let V be the volume of the solid generated when T is rotated about the line x = a, where a 1. Express a in terms of V.45E 46E Use cylindrical shells to find the volume of the solid. A right circular cone with height h and base radius r Use cylindrical shells to find the volume of the solid. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure. (a) Guess which ring has more wood in it. (b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius r through the center of a sphere of radius R and express the answer in terms of h.A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds How much work is done when a hoist lifts a 200-kg rock to a height of 3 m?3E When a particle is located a distance x meters from the origin, a force of cos(x/3) newtons acts on it. How much work is done in moving the particle from x = 1 to x = 2? Interpret your answer by considering the work done from x = 1 to x = 1.5 and from x = 1.5 to x = 2.Shown is the graph of a force function (in newtons) that increases to its maximum value and then remains constant. How much work is done by the force in moving an object a distance of 8 m?6E A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?A spring has a natural length of 40 cm. If a 60-N force is required to keep the spring compressed 10 cm, how much work is done during this compression? How much work is required to compress the spring to a length of 25 cm?Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm. (a) How much work is needed to stretch the spring from 35 cm to 40 cm? (b) How far beyond its natural length will a force of 30 N keep the spring stretched?If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?A spring has natural length 20 cm. Compare the work W1 done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related?If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (a) How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling half the rope to the top of the building?Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A thick cable, 60 ft long and weighing 180 lb, hangs from a winch on a crane. Compute in two different ways the work done if the winch winds up 25 ft of the cable. (a) Follow the method of Example 4. (b) Write a function for the weight of the remaining cable after x feet has been wound up by the winch. Estimate the amount of work done when the winch pulls up x ft of cable.Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of 12m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that its level with the upper end.Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A circular swimming pool has a diameter of 24ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5 lb/ft3.)Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. An aquarium 2m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg/ m3.)Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A spherical water tank, 24 ft in diameter, sits atop a 60 ft tower. The tank is filled by a hose attached to the bottom of the sphere. If a 1.5 horsepower pump is used to deliver water up to the tank, how long will it take to fill the tank? (One horsepower = 550 ft-lb of work per second.)A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.Suppose that for the tank in Exercise 23 the pump breaks down after 4. 7 105 J of work has been done. What is the depth of the water remaining in the tank?Solve Exercise 24 if the tank is half full of oil that has a density of 900 kg/m3 When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P(V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = r2P. Show that the work done by the gas when the volume expands from volume V1 to volume V2 is W=V1V2PdV In a steam engine the pressure P and volume V of steam satisfy the equation PV1.4 = k, where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use Exercise 29 to calculate the work done by the engine during a cycle when the steam starts at a pressure of 160 lb/in2 and a volume of 100 in3 and expands to a volume of 800 in3.31E 32E (a) Newtons Law of Gravitation states that two bodies with masses m1, and m2, attract each other with a force F=Gm1m2r2 where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r = a to r = b. (b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earths mass is 5.98 1024 kg and is concentrated at its center. Take the radius of the earth to be 6.37 106 m and G = 6.67 1011 Nm2/kg2 The Great Pyramid of King Khufu was built of limestone in Egypt over a 20-year time period from 2580 bc to 2560 bc. Its base is a square with side length 756 ft and its height when built was 481 ft. (It was the tallest man-made structure in the world for more than 3800 years.) The density of the limestone is about 150 lb/ft3. (a) Estimate the total work done in building the pyramid. (b) If each laborer worked 10 hours a day for 20 years, for 340 days a year, and did 200 ft-lb/h of work in lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?Find the average value of the function on the given interval. f(x) = 3x2 + 8x, [1, 2]Find the average value of the function on the given interval. f(x)=x,[0,4]Find the average value of the function on the given interval. g(x) = 3 cos x, [/2, /2]Find the average value of the function on the given interval. g(t)=t3+t2,[1,3]Find the average value of the function on the given interval. f(t) = esin t cos t, [0, /2]Find the average value of the function on the given interval. f(x) = x2/(x3 + 3)2, [1, 1]Find the average value of the function on the given interval. h(x) = cos4x sin x, [0, ]Find the average value of the function on the given interval. h(u) = (ln u)/u, [1, 5](a) Find the average value of f on the given interval. (b) Find c such that fave = f(c). (c) Sketch the graph off and a rectangle whose area is the same as the area under the graph of f. f(x) = (x 3)2, [2, 5](a) Find the average value of f on the given interval. (b) Find c such that fave = f(c). (c) Sketch the graph off and a rectangle whose area is the same as the area under the graph of f. f(x) = 1/x, [1, 3](a) Find the average value of f on the given interval. (b) Find c such that fave = f(c). (c) Sketch the graph off and a rectangle whose area is the same as the area under the graph of f. f(x) = 2 sin x sin 2x, [0, ]12E If f is continuous and 13f(x)dx=8, show that f takes on the value 4 at least once on the interval [1, 3].Find the numbers b such that the average value of f(x) = 2 + 6x 3x2 on the interval [0, b] is equal to 3.Find the average value of f on [0, 8].The velocity graph of an accelerating car is shown. (a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity?In a certain city the temperature (in F) t hours after 9 am was modeled by the function T(t)=50+14sint12 Find the average temperature during the period from 9 am to 9 pm.The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r)=P4l(R2r2) where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 3.7. 7). Find the average velocity (with respect to r) over the interval 0 r R. Compare the average velocity with the maximum velocity.The linear density in a rod 8 m long is 12/x+1kg/m, where x is measured in meters from one end of the rod. Find the average density of the rod.(a) A cup of coffee has temperature 95C and takes 30 minutes to cool to 61C in a room with temperature 20C. Use Newtons Law of Cooling (Section 3.8) to show that the temperature of the coffee after t minutes is T(t)=20+75ekt where k 0.02. (b) What is the average temperature of the coffee during the first half hour?21E 22E Use the result of Exercise 5.5.83 to compute the average volume of inhaled air in the lungs in one respiratory cycle.Use the diagram to show that if f is concave upward on [a, b], then favef(a+b2)25E 26E (a) Draw two typical curves y = f(x) and y = g(x), where f(x) g(x) for a x b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area. (b) Explain how the situation changes if the curves have equations x = f(y) and x = g(y), where f(y) g(y) for c y d.Suppose that Sue runs faster than Kathy throughout a 1500-meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?3RCC 4RCC Suppose that you push a book across a 6-meter-long table by exerting a force f(x) at each point from x = 0 to x = 6. What does 06f(x)dx represent? If f(x) is measured in newtons, what are the units for the integral?6RCC Find the area of the region bounded by the given curves. y = x2, y = 4x x2 Find the area of the region bounded by the given curves. y=x, y=x3, y = x 2 3RE Find the area of the region bounded by the given curves. x + y = 0, x = y2 + 3y Find the area of the region bounded by the given curves. y = sin(x/2), y = x2 2x Find the area of the region bounded by the given curves. y=x, y = x2, x = 2 7RE Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 1 + y2, y = x 3; about the y-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x = 9 y2; about x = 1 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x2 + 1, y = 9 x2; about y = 1 11RE Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = tan x, y = x, x = /3; about the y-axis 13RE Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=x, y = x2; about y = 2 Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines. (a) The x-axis (b) The y-axis (c) y = 2 Let be the region in the first quadrant bounded by the curves y = x3 and y = 2x x2. Calculate the following quantities. (a) The area of (b) The volume obtained by rotating about the x-axis (c) The volume obtained by rotating about the y-axis 17RE Let be the region bounded by the curves y = 1 x2 and y = x6 x + 1. Estimate the following quantities. (a) The x-coordinates of the points of intersection of the curves (b) The area of (c) The volume generated when is rotated about the x-axis (d) The volume generated when is rotated about the y-axis 19RE Each integral represents the volume of a solid. Describe the solid. 0/22cos2xdx Each integral represents the volume of a solid. Describe the solid. 0(2sinx)2dx Each integral represents the volume of a solid. Describe the solid. 042(6y)(4yy2)dy The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.The base of a solid is the region bounded by the parabolas y = x2 and y = 2 x2. Find the volume of the solid if the cross-sections perpendicular to the x-axis are squares with one side lying along the base.25RE 26RE 27RE A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank. (b) After 4000 ft-lb of work has been done, what is the depth of the water remaining in the tank?A steel tank has the shape of a circular cylinder oriented vertically with diameter 4 m and height 5 m. The tank is currently filled to a level of 3 m with cooking oil that has a density of 920 kg/m3. Compute the work required to pump the oil out through a 1-m spout at the top of the tank.31RE 32RE 33RE 34RE 1P There is a line through the origin that divides the region bounded by the parabola y = x x2 and the x-axis into two regions with equal area. What is the slope of that line?The figure shows a horizontal line y = c intersecting the curve y = 8x 27x3. Find the number c such that the areas of the shaded regions are equal. FIGURE FOR PROBLEM 3 A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base. (a) Determine a way to slice the water into parallel rectangular cross-sections and then set up a definite integral for the volume of the water in the glass. (b) Determine a way to slice the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water. (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b). (d) Find the volume of the water in the glass from purely geometric considerations. (e) Suppose the glass is tilted until the water exactly covers half the base. In what direction can you slice the water into triangular cross-sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.(a) Show that the volume of a segment of height h of a sphere of radius r is V=13h2(3rh) (See the figure.) (b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of one segment is twice the volume of the other, then x is a solution of the equation 3x3 9x + 2 = 0 where 0 x 1. Use Newtons method to find x accurate to four decimal places. (c) Using the formula for the volume of a segment of a sphere, it can be shown that the depth x to which a floating sphere of radius r s inks in water is a root of the equation x3 3rx2 + 4r3s = 0 where s is the specific gravity of the sphere. Suppose a wooden sphere of radius 0.5 m has specific gravity 0.75. Calculate, to four-decimal-place accuracy, the depth to which the sphere will sink. (d) A hemispherical bowl has radius 5 inches and water is running into the bowl at the rate of 0.2 in3/s. (i) How fast is the water level in the bowl rising at the instant the water is 3 inches deep? (ii) At a certain instant, the water is 4 inches deep. How long will it take to fill the bowl? FIGURE FOR PROBLEM 5 Archimedes Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density 0, floating partly submerged in a fluid of density f, the buoyant force is given by F=fgh0A(y)dy, where g is the acceleration due to gravity and A(y) is the area of a typical cross-section of the object (see the figure). The weight of the object is given by W=0ghLhA(y)dy (a) Show that the percentage of the volume of the object above the surface of the liquid is 100f0f (b) The density of ice is 917 kg/m3 and the density of seawater is 1030 kg/m3. What percentage of the volume of an iceberg is above water? (c) An ice cube floats in a glass filled to the brim with water. Does the water overflow when the ice melts? (d) A sphere of radius 0.4 m and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the water is 1000 kg/m3. FIGURE FOR PROBLEM 6 7P 8P The figure shows a curve C with the property that, for every point P on the middle curve y = 2x2, the areas A and B are equal. Find an equation for C. FIGURE FOR PROBLEM 9 A paper drinking cup filled with water has the shape of a cone with height h and semi-vertical angle . (See the figure.) A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The clock is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x = f(y) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of f about the y-axis. Let V denote the volume of water and h the height of the water level at time t. (a) Determine V as a function of h. (b) Show that dVdt=[f(h)]2dhdt (c) Suppose that A is the area of the hole in the bottom of the container. It follows from Torricellis Law that the rate of change of the volume of the water is given by dVdt=kAh where k is a negative constant. Determine a formula for the function f such that dh/dt is a constant C. What is the advantage in having dh/dt = C? FIGURE FOR PROBLEM 11 A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed , then the container will induce a rotational motion in the liquid around the same axis. Eventually, the liquid will be rotating at the same angular speed as the container. The surface of the liquid will be convex, as indicated in the figure, because the centrifugal force on the liquid particles increases with the distance from the axis of the container. It can be shown that the surface of the liquid is a paraboloid of revolution generated by rotating the parabola y=h+2x22g about the y-axis, where g is the acceleration due to gravity. (a) Determine h as a function of . (b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top? (c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis. (i) Determine the angular speed of the container and the volume of the fluid. (ii) How far below the top of the tank is the liquid at the wall of the container? FIGURE FOR PROBLEM 12 13P If the tangent at a point P on the curve y = x3 intersects the curve again at Q, let A be the area of the region bounded by the curve and the line segment PQ. Let B be the area of the region defined in the same way starting with Q instead of P. What is the relationship between A and B?Evaluate the integral using integration by parts with the indicated choices of u and dv. 1. xe2xdx; u = x, dv = e2x dx Evaluate the integral using integration by parts with the indicated choices of u and dv. 2. xlnxdx; u = ln x, dv=xdx Evaluate the integral. 3. xcos5xdx Evaluate the integral. 4.ye0.2ydy Evaluate the integral. 5. te3tdt Evaluate the integral. 6. (x1)sinxdx Evaluate the integral. 7. (x2+2x)cosxdx Evaluate the integral. 8. t2sintdt Evaluate the integral. 9. cos1xdx Evaluate the integral. 10. lnxdx Evaluate the integral. 11. t4lntdt Evaluate the integral. 12. tan12ydy Evaluate the integral. 13. tcsc2tdt Evaluate the integral. 14. xcoshaxdx Evaluate the integral. 15. (lnx)2dx Evaluate the integral. 16. z10zdz Evaluate the integral. 17. e2sin3d Evaluate the integral. 18. ecos2d Evaluate the integral. 19. z3ezdz Evaluate the integral. 20. xtan2xdx Evaluate the integral. 21. xe2x(1+2x)2dx Evaluate the integral. 22. (arcsinx)2dx Evaluate the integral. 23. 01/2xcosxdx Evaluate the integral. 24. 01(x2+1)exdx Evaluate the integral. 25. 02ysinhydy Evaluate the integral. 26. 12w2lnwdw Evaluate the integral. 27. 15lnRR2dR Evaluate the integral. 28. 02t2sin2tdt Evaluate the integral. 29. 0xsinxcosxdx Evaluate the integral. 30. 13arctan(1/x)dx Evaluate the integral. 31. 15MeMdM Evaluate the integral. 32. 12(lnx)2x3dx Evaluate the integral. 33. 0/3sinxln(cosx)dx Evaluate the integral. 34. 01r34+r2dr Evaluate the integral. 35. 12x4(lnx)2dx Evaluate the integral. 36. 0tessin(ts)ds First make a substitution and then use integration by parts to evaluate the integral. 37. exdx First make a substitution and then use integration by parts to evaluate the integral. 38. cos(lnx)dx First make a substitution and then use integration by parts to evaluate the integral. 39. /23cos(2)d First make a substitution and then use integration by parts to evaluate the integral. 40. 0ecostsin2tdt First make a substitution and then use integration by parts to evaluate the integral. 41 xln(1+x)dx First make a substitution and then use integration by parts to evaluate the integral. 42. arcsin(lnx)xdx Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 43. xe2xdx Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 44. x3/2lnxdx Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 45. x31+x2dx Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). 46. x2sin2xdx (a) Use the reduction formula in Example 6 to show that sin2xdx=x2sin2x4+C (b) Use part (a) and the reduction formula to evaluate sin4xdx.(a) Prove the reduction formula cosnxdx=1ncosn1xsinx+n1ncosn2xdx (b) Use part (a) to evaluate cos2xdx. (c) Use parts (a) and (b) to evaluate cos4xdx.(a) Use the reduction formula in Example 6 to show that 0/2sinnxdx=n1n0/2sinn2xdx where n 2 is an integer. (b) Use part (a) to evaluate 0/2sin3xdx and 0/2sin5xdx. (c) Use part (a) to show that, for odd powers of sine, 0/2sin2n+1xdx=2462n357(2n+1)Prove that, for even powers of sine, 0/2sin2nxdx=135(2n1)2462n2 Use integration by parts to prove the reduction formula. 51. (lnx)ndx=x(lnx)nn(lnx)n1dx Use integration by parts to prove the reduction formula. 52. xnexdx=xnexnxn1exdx Use integration by parts to prove the reduction formula. 53. tannxdx=tann1xn1tann2xdx(n1)Use integration by parts to prove the reduction formula. 54. secnxdx=tanxsecn2xn1+n2n1secn2xdx(n1)Use Exercise 51 to find (lnx)3dx.Use Exercise 52 to find x4exdx.Find the area of the region bounded by the given curves. 57. y = x2 ln x, y = 4 ln x Find the area of the region bounded by the given curves. 58. y = x2ex, y = xex Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. 59. y=arcsin(12x), y = 2 x2 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. 60. y = x ln(x + 1), y = 3x x2 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. 61. y = cos(x/2), y = 0, 0 x 1; about the y-axis Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. 62. y = ex, y = ex, x = 1; about the y-axis Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. 63. y = ex, y = 0, x = 1, x = 0; about x = 1 64E Calculate the volume generated by rotating the region bounded by the curves y = ln x, y = 0, and x = 2 about each axis. (a) The y-axis (b) The x-axis Calculate the average value of f(x) = x sec2x on the interval [0, /4].The Fresnel function S(x)=0xsin(12t2)dt was discussed in Example 5.3.3 and is used extensively in the theory of optics. FindS(x)dx. [Your answer will involve S(x).]A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ve (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation v(t)=gtvelnmrtm where g is the acceleration due to gravity and t is not too large. If g = 9.8 m/s2, m = 30,000 kg, r = 160 kg/s, and ve = 3000 m/s, find the height of the rocket one minute after liftoff.A particle that moves along a straight line has velocity v(t) = t2et meters per second after t seconds. How far will it travel during the first t seconds?70E Suppose that f(l) = 2, f(4) = 7, f(1) = 5, f(4) = 3, and f" is continuous. Find the value of 14xf(x)dx.(a) Use integration by parts to show that f(x)dx=xf(x)xf(x)dx (b) If f and g are inverse functions and f is continuous, prove that abf(x)dx=bf(b)af(a)f(a)f(b)g(y)dy [Hint: Use part (a) and make the substitution y = f(x).] (c) In the case where f and g are positive functions and b a 0, draw a diagram to give a geometric interpretation of part (b). (d) Use part (b) to evaluate 1elnxdx.We arrived at Formula 6.3.2, V=ab2xf(x)dx, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where f is one-to-one and therefore has an inverse function g. Use the figure to show that V=b2da2ccd[g(y)]2dy Make the substitution y = f(x) and then use integration by parts on the resulting integral to prove that V=ab2xf(x)dx Let In=0/2sinnxdx. (a) Show that I2n+2 I2n+1 I2n. (b) Use Exercise 50 to show that I2n+2I2n=2n+12n+2 (c) Use parts (a) and (b) to show that 2n+12n+2I2n+2I2n1 and deduce that limn I2n+1/I2n = 1. (d) Use part (c) and Exercises 49 and 50 to show that limn2123434565672n2n12n2n+1=2 This formula is usually written as an infinite product: 2=212343456567 and is called the Wallis product. (e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.Evaluate the integral. 1. sin2xcos3xdx Evaluate the integral. 2. sin3cos4d Evaluate the integral. 3. 0/2sin7cos5d Evaluate the integral. 4. 0/2sin5xdx Evaluate the integral. 5. sin5(2t)cos2(2t)dt Evaluate the integral. 6. tcos5(t2)dt Evaluate the integral. 7. 0/2cos2d Evaluate the integral. 8. 02sin2(13)d Evaluate the integral. 9. 0cos4(2t)dt Evaluate the integral. 10. 0sin2tcos4tdt Evaluate the integral. 11. 0/2sin2xcos2xdx Evaluate the integral. 12. 0/2(2sin)2d Evaluate the integral. 13. cossin3d Evaluate the integral. 14. sin2(1/t)t2dt Evaluate the integral. 15. cotxcos2xdx Evaluate the integral. 16. tan2xcos3xdx

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