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All Textbook Solutions for Calculus
Water Tower A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)Minimum Volume The function y=4(x2/4) on the interval [0.4] is revolved about the line y=b (sec figure). (a) Find the volume of the resulting solid as a function of b. (b) Use a graphing utility to graph the function in part (a), and use the graph to approximate the value of b that minimizes the volume of the solid. (c) Use calculus to find the value of b that minimizes the volume of the solid, and compare the result with the answer to part (b).68E69E70E71E72E73E74E75E76E1EFinding the Volume of a Solid In Exercises 3-12, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y=1xFinding the Volume of a Solid In Exercises 3-12, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y=x4E5E6E7E8E9E10EFinding the Volume of a Solid In Exercises 114, use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. y=x=2,y=0,x=412E13E14E15EFinding the Volume of a Solid In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y=1xFinding the Volume of a Solid In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y=1xFinding the Volume of a Solid In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. x+y2=419E20E21EFinding the Volume of a Solid In Exercises 13-22, use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y=x+2,y=x,y=023EFinding the Volume of a Solid In Exercises 23-26, use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given line. y=x,y=0,x=4,aboutthelinex=625EFinding the Volume of a Solid In Exercises 23-26, use the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given line. y=13x3,y=6xx2,aboutthelinex=327E28E29EChoosing a Method In Exercises 29-32, use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. y=10x2,y=0,x=1,x=5 (a) the x-axis (b) the y-axis (c) the line y=1031E32E33EFinding the Volume of a Solid Using Technology In Exercises 33-36, (a) use a graphing utility to graph the region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis. y=1x3,y=0,x=035E36E38E37E39EEXPLORING CONCEPTS Comparing Integrals In Exercises 39 and 40, give a geometric argument that explains why the integrals have equal values. 02[16(2y)2]dy=204y(x2)dxComparing Volumes The region in the figure is revolved about the indicated axes and line. Order the volumes of the resulting solids from least to greatest. Explain your reasoning. (a) x-axis (b) y-axis (c) x=442E43E44E45E46EMachine Part A solid is generated by revolving the region bounded by y=12x2 and y=2 about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-fourth of the volume is removed. Find the diameter of the hole.Machine Part A solid is generated by revolving the region bounded by y=9x2 - and y=0 about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole.Volume of a Torus A torus is formed by revolving the region bounded by the circle x2+y2=1 about the line x=2 (see figure). Find the volume of this doughnut-shaped solid. (Hint: The integral 111x2 dx represents the area of a semicircle.)Volume of a Torus Repeat Exercise 49 for a torus formed by revolving the region bounded by the circle x2+y2=r2 about the line x=R, where rR.51EFinding Volumes of Solids (a) Use differentiation to verify that xcosxdx=cosx+xsinx+C (b) Use the result of part (a) to find the volume of the solid generated by revolving each plane region about the y-axis. (Hint: Begin by approximating the points of intersection.)Volume of a Segment of a Sphere Let a sphere of radius r be cut by a plane, thereby forming a segment of height h. Show that the volume of this segment is 13h2(3rh).Volume of an Ellipsoid Consider the plane region bounded by the graph of the ellipse (xa)2+(yb)2=1 where a0 and b0. Show that the volume of the ellipsoid formed when this region is revolved about the y-axis is 43a2b. What is the volume when the region is revolved about the x-axis?Exploration Consider the region bounded by the graphs of y=axn,y=abn, and x=0, as shown in the figure. (a) Find the ratio R1(n) of the area of the region to the area of the circumscribed rectangle. (b) Find limnR1(n) and compare the result with the area of the circumscribed rectangle. (c) Find the volume of the solid of revolution formed by revolving the region about the y-axis. Find the ratio R2(n) of this volume to the volume of the circumscribed right circular cylinder. (d) Find limnR2(n) and compare the result with the volume of the circumscribed cylinder. (e) Use the results of parts (b) and (d) to make a conjecture about the shape of the graph of y=axn,0xb, as n.Think About It Match each integral with the solid whose volume it represents and give the dimensions of each solid. (a) Right circular cone (b) Torus (c) Sphere (d) Right circular cylinder (e) Ellipsoid (i) 20rhxdx (ii) 20rhx(1xr)dx (iii) 20r2xr2x2dx (iv) 20r2ax1x2b2dx (v) 2rr(Rx)(2r2x2)dxVolume of a Storage Shed A storage shed has a circular base of diameter 80 feet. Starting at the center, the interior height is measured every 10 feet and recorded in the table (see figure). Find the volume of the shed. x Height 0 50 10 45 20 40 30 20 40 058E59E60EFinding Volumes of Solids Consider the graph of y2=x(4x)2, as shown in the figure. Find the volumes of the solids that are generated when the loop of this graph is revolved about (a) the x-axis, (b) the y-axis, and (c) the line x=4.1E2E3EFinding Arc Length In Exercises 316, find the arc length of the graph of the function over the indicated interval. y=x36+12x5E6EFinding Arc Length In Exercises 7-20, find the arc length of the graph of the function over the indicated interval. y=32x3/2,[1,8]8E9E10E11E12EFinding Arc Length In Exercises 7-20, find the arc length of the graph of the function over the indicated interval. y=12(ex+ex),[0,2]14EFinding Arc Length In Exercises 7-20, find the arc length of the graph of the function over the indicated interval. x=13(y2+2)3/2,0y4Finding Arc Length In Exercises 7-20, find the arc length of the graph of the function over the indicated interval. x=13y(y3),1y417E18E19E20E21E22EFinding Arc Length In Exercises 21-30, (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. x=ey,0y224EFinding Arc Length In Exercises 21-30, (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) write a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. y=2arctanx,[0,1]26E27E28E29E30ELength of a Catenary Electrical wires suspended between two towers form a catenary (see figure) modeled by the equation y=20coshx20,20x20 where x and y are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable.Roof Area A barn is 100 feet long and 40 feet wide (see figure). A cross section of the roof is the inverted catenary y=3110(ex/20+ex/20). Find the number of square feet of roofing on the bam.Length of Gateway Arch The Gateway Arch in St. Louis, Missouri, is closely approximated by the inverted catenary y=693.859768.7672cosh0.0100333x,299.2239x299.239. Use the integration capabilities of a graphing utility to approximate the length of this curve (see figure).Astroid Find the total length of the graph of the astroid x2/3+y2/3=435E36EFinding the Area of a Surface of Revolution In Exercises 39-44, write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y=13x3Finding the Area of a Surface of Revolution In Exercises 39-44, write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y=2xFinding the Area of a Surface of Revolution In Exercises 39-44, write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y=x36=12x,1x2Finding the Area of a Surface of Revolution In Exercises 39-44, write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y=3x,0x341E42EFinding the Area of a Surface of Revolution In Exercises 45-48, write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the y-axis. y=x3+2,1x844E45E46E47E48E49EPrecalculus and Calculus What precalculus formula and representative element are used to develop the integration formula for are length?51E52E53E54E55E56EUsing a Sphere Find the area of the segment of a sphere formed by revolving the graph of y=9x2,0x2, about the y-axis.58EModeling Data The circumference C (in inches) of a vase is measured at three-inch intervals starting at its base. The measurements are shown in the table, where y is the vertical distance in inches from the base. y 0 3 6 9 12 15 18 C 50 65.5 70 66 58 51 48 (a) Use the data to approximate the volume of the vase by summing the volumes of approximating disks. (b) Use the data to approximate the outside surface area (excluding the base) of the vase by summing the outside surface areas of approximating frustums of right circular cones. (c) Use the regression capabilities of a graphing utility to find a cubic model for the points (y, r), where r=C/(2). Use the graphing utility to plot the points and graph the model. (d) Use the model in part (c) and the integration capabilities of a graphing utility to approximate the volume and outside surface area of the vase. Compare the results with your answers in parts (a) and (b).Modeling Data Property bounded by two perpendicular roads and a stream is shown in the figure. All distances are measured in feet. (a) Use the regression capabilities of a graphing utility to fit a fourth-degree polynomial to the path of the stream. (b) Use the model in part (a) to approximate the area of the property in acres. (c) Use the integration capabilities of a graphing utility to find the length of the stream that bounds the property.61E62EApproximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.) Length of Pursuit A fleeing object leaves the origin and moves up the y-axis (see figure). At the same time, a pursuer leaves the point (1,0) and always moves toward the fleeing object. The pursuers speed is twice that of the fleeing object. The equation of the path is modeled by y=13(x3/23x1/2+2) How far has the fleeing object traveled when it is caught? Show that the pursuer has traveled twice as far.Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.) Bulb Design An ornamental light bulb is designed by revolving the graph of y=13x1/2x3/2,0x13 about the x-axis, where x and y are measured in feet (see figure). Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. Assume that the thickness of the glass is 0.015 inch.Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.) Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of x2/3+y2/3=4,0y8, about the y-axis.Approximating Arc Length or Surface Area In Exercises 65-68, write the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface area. (You will learn how to evaluate this type of integral in Section 8.8.) Using a Loop Consider the graph of y2=112x(4x)2 shown in the figure. Find the area of the surface formed when the loop of this graph is revolved about the x-axis.Suspension Bridge A cable for a suspension bridge has shape of a parabola with equation y=kx2. Let h represent height of the cable from its lowest point to its highest point let 2w represent the total span of the bridge (see figure). Show that the length C of the cable is given by C=20w1+(4h2w2)x2dx68E69E70EConstant Force In Exercises 5-8, determine the work done by the constant force. A 1200-pound steel beam is lifted 40 feet.Constant Force In Exercises 14, determine the work done by the constant force. An electric hoist lifts a 2500pound car 6 feet.Constant Force In Exercises 5-8, determine the work done by the constant force. A force of 112 newtons is required to slide a cement block 8 meters in a construction project.Constant Force In Exercises 14, determine the work done by the constant force. The locomotive of a freight train pulls its cars with a constant force of 9 tons a distance of one-half mile.Hooke's Law In Exercises 9-14, use Hookes Law to determine the work done by the variable force in the spring problem. A force of 5 pounds compresses a 15-inch spring a total of 3 inches. How much work is done in compressing the spring 7 inches?Hooke's Law In Exercises 9-14, use Hookes Law to determine the work done by the variable force in the spring problem. A force of 250 newtons stretches a spring 30 centimeters. How much work is done in stretching the spring from 20 centimeters to 50 centimeters?Hooke's Law In Exercises 9-14, use Hookes Law to determine the work done by the variable force in the spring problem. A force of 20 pounds stretches a spring 9 inches in an exercise machine. Find the work done in stretching the spring 1 foot from its natural position.Hooke's Law In Exercises 9-14, use Hookes Law to determine the work done by the variable force in the spring problem. An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the work done by the pair of springs.9E10EPropulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-metric-ton satellite to a height of (a) 100 miles above Earth and (b) 300 miles above Earth.Propulsion Use the information in Exercise 15 to write the work W of the propulsion system as a function of the height h of the satellite above Earth. Find the limit (if it exists) of W as h approaches infinity.Propulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 10-metric-ton satellite to a height of (a) 11,000 miles above Earth and (b) 22,000 miles above Earth.Propulsion A lunar module weighs 12 metric tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.Pumping Water A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (sec figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty (a) half of the tank and (b) all of the tank?16E17E18E19E20E21E22EPumping Gasoline In Exercises 27 and 28, find the work done in pumping gasoline that weighs 42 pounds per cubic foot. A cylindrical gasoline tank 3 feet in diameter and 4 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire contents of the fuel tank into the tractor.Pumping Gasoline In Exercises 27 and 28, find the work done in pumping gasoline that weighs 42 pounds per cubic foot. The top of a cylindrical gasoline storage tank at a service station is 4 feet below ground level. The axis of the tank is horizontal and its diameter and length are 5 feet and 12 feet, respectively. Find the work done in pumping the entire contents of the full tank to a height of 3 feet above ground level.Winding a Chain In Exercises 29-32, consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Kind the work done by the winch in winding up the specified amount of chain. Wind up the entire chain.Winding a Chain In Exercises 29-32, consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Kind the work done by the winch in winding up the specified amount of chain. Wind up one-third of the chain.27E28ELifting a Chain In Exercises 33 and 34, consider a 15-foot hanging chain that weighs 3 pounds per foot. Find the work done in lifting the chain vertically to the indicated position. Take the bottom of the chain and raise it to the 15-foot level, leaving the chain doubled and still hanging vertically (see figure).Lifting a Chain In Exercises 33 and 34, consider a 15-foot hanging chain that weighs 3 pounds per foot. Find the work done in lifting the chain vertically to the indicated position. Repeat Exercise 33 raising the bottom of the chain to the 12-foot level.31E32E33EHOW DO YOU SEE IT? The graphs show the force Fi (in pounds) required to move an object 9 feet along the x-axis. Order the force functions from the one that yields the least work to the one that yields the most work without doing any calculations. Explain your reasoning.Electric Force Two electrons repel each other with a force that varies inversely as the square of the distance between them. One electron is fixed at the point (2,4). Find the work done in moving the second electron from (- 2.4) to (1,4).35E37E38E39E40E41EHydraulic Press In Exercises 45-48, use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force F (in pounds) and the distance x (in feet) the press moves is given. Force Interval F(x)=1000sinhx 0x2Modeling Data The hydraulic cylinder on a woodsplitter has a 4-inch bore (diameter) and a stroke of 2 feet. The hydraulic pump creates a maximum pressure of 2000 pounds per square inch. Therefore, the maximum force created by the cylinder is 2000(22)=8000 pounds. (a) Find the work done through one extension of the cylinder, given that the maximum force is required. (b) The force exerted in splitting a piece of wood is variable. Measurements of the force obtained in splitting a piece of wood are shown in the table. The variable x measures the extension of the cylinder in feet, and F is the force in pounds. Use the regression capabilities of a graphing utility to find a fourth-degree polynomial model for the data. Plot the data and graph the model. x 0 13 23 1 43 53 2 F(x) 0 20,000 22,000 15,000 10,000 5000 0 (c) Use the model in part (b) to approximate the extension of the cylinder when the force is maximum. (d) Use the model in part (b) to approximate the work done in splitting the piece of wood.43E1E2E3E4E5E6E7E8E9E10ECenter of Mass of a Two-Dimensional System In Exercises 11-14, find the center of mass of the given system of point masses. mi. 12 6 4.5 15 (xi,yi) (2,3) (-1,5) (6.8) (2, -2)12E13E14E15ECenter of Mass of a Planar Lamina In Exercises 1326, find Mx, My, and (x, y) for the laminas of uniform density p bounded by the graphs of the equations. y=12x2,y=0,x=217E18E19E20E21E22E23ECenter of Mass of a Planar Lamina In Exercises 15-28, find Mx,My , and (x, y) for the lamina of uniform density p bounded by the graphs of the equations. x=3yy2,x=025E26E27E28E29E30E31E32E33E34EFinding the Center of Mass Find the center of mass of the lamina in Exercise 31 when the circular portion of the lamina has twice the density of the square portion of the lamina.36E37E38E39E40E41EPlanar Lamina What is a planar lamina? Describe what is meant by the center of mass (x,y) of a planar lamina.44E45E46ECentroid of a Common Region In Exercises 45-50, find and/or verify the centroid of the common region used in engineering. Trapezoid Find the centroid of the trapezoid with vertices (0,0), (0, a), (c, b), and (c, 0). Show that it is the intersection of the line connecting the midpoints of the parallel sides and the line connecting the extended parallel sides, as shown in the figure.48ECentroid of a Common Region In Exercises 45-50, find and/or verify the centroid of the common region used in engineering. Semiellipse Find the centroid of the region bounded by the graph of y=baa2x2 and y=0 (see figure).50E51EGraphical and Numerical Reasoning Consider the region bounded by the graphs of y=x2n and y=b, where b0 and n is a positive integer. (a) Sketch a graph of the region. (b) Set up the integral for finding My. Because of the form of the integrand, the value of the integral can be obtained without integrating. What is the form of the integrand? What is the value of the integral and what is the value of x? (c) Use the graph in part (a) to determine whether yb2 or yb2- Explain. (d) Use integration to find y as a function of n. (e) Use the result of part (d) to complete the table. n 1 2 3 4 y (f) Hind limny (g) Give a geometric explanation of the result in part (f).53EModeling Data The manufacturer of a boat needs approximate the center of mass of a section of the hull, coordinate system is superimposed on a prototype (see figure). The measurements (in feet) for the right half of the symmetric prototype are listed in the table. x 0 0.5 1.0 1.5 2 h 1.50 1.45 1.30 0.99 0 d 0.50 0.48 0.43 0.33 0 (a) Use the regression capabilities of a graphing utility to find fourth-degree polynomial models for both curves shown in the figure. (b) Use the integration capabilities of a graphing utility and the models to approximate the center of mass of the hull section.55E56E57E58EForce on a Submerged Sheet In Exercises 3-6, the area of the top side of a piece of sheet metal is given. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. 3 square feet2E3E4EBuoyant Force In Exercises 5 and 6, find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid.6EFluid Force on a Tank Wall In Exercises 9-14, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Rectangle8EFluid Force on a Tank Wall In Exercises 914, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. TrapezoidFluid Force on a Tank Wall In Exercises 914, find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Semicircle11E12EFluid Force of a Water In Exercises 15-18, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 network per cubic meter. SquareFluid Force of Water In Exercises 15-18, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter. RectangleFluid Force of Water In Exercises 1518, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meters. SquareFluid Force of Water In Exercises 1518, find the fluid force on the vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meters. Triangle17EForce on a Concrete formIn Exercises 1922, the figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. Semiellipse, y=3416x219E20EFluid Force of Gasoline A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank when the tank is half full, where the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.22E33E31EHOW DO YOU SEE IT? Two identical semicircular windows are placed at the same depth in the vertical wall of an aquarium (see figure). Which is subjected to the greater fluid force? Explain.Fluid Force on a Circular Plate A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs w pounds per cubic foot. The center of the circle is k feet below the surface of the fluid, where kr. Show that the fluid force on the surface of the plate is F=wk(r2).Fluid Force on a Circular Plate Use the result of Exercise 29 to find the fluid force on the circular plate shown in each figure. Assume that the tank is filled with water and the measurements are given in feet.25EFluid Force on a Rectangular Plate Use the result of Exercise 31 to find the fluid force on the rectangular plate shown in each figure. Assume that the tank is filled with water and the measurements are given in feet.Submarine Porthole A square porthole on a vertical side of a submarine (submerged in seawater) has an area of 1 square foot. Find the fluid force on the porthole, assuming that the center of the square is 15 feet below the surface.Submarine Pothole Repeat Exercise 33 for a circular porthole that has a diameter of 1 foot. The center of the circle is 15 feet below the surface.29E30EFinding the Area of a RegionIn Exercises 110, sketch the region bounded by the graphs of the equations and find the area of the region. y=612x2,y=34x,x=2,x=2Finding the Area of a RegionIn Exercises 1-10, sketch the region bounded by the graphs of the equations and find the area of the region. y=1x2,y=4,x=53RE4RE5RE6RE7RE8REFinding the Area of a Region In Exercises 1-10, sketch the region bounded by the graphs of the equations and find the area of the region. y=sinx,ycosx,4x5410REFinding the Area of a Region In Exercises 11-14, (a) use a graphing utility to graph the region bounded by the graphs of the equations and (b) use the integration capabilities of the graphing utility to approximate the area of the region to four decimal places. y=x28x+3y=3+8xx212REFinding the Area of a RegionIn Exercises 11-14, (a) use a graphing utility to graph the region bounded by the graphs of the equations and (b) use the integration capabilities of the graphing utility to approximate the area of the region to four decimal places. x+y=1,y=0,x=014RENumerical Integration Estimate the surface area of the pond using (a) the Trapezoidal Rule and (b) Simpsons Rule.16RE19RE22RE20RE21REFinding the Volume of a Solid In Exercises 23 and 24, use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. y=x,y=2,x=0 (a) the x-axis (b) the line y-axis (c) the line x=3 (d) the line x=6Finding the Volume of a Solid In Exercises 23 and 24, use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. y=x,y=2,x=0 (a) the x-axis (b) the line y=2 (c) the y-axis (d) the line x=1Gasoline Tank A gasoline tank is an oblate spheroid generated by revolving the region bounded by the graph of x216+y29=1 about the y-axis, where x and y are measured in feet. How much gasoline can the tank hold?Using Cross Sections Find the volume of the solid whose base is bounded by the circle x2+y2=9 and whose cross sections perpendicular to the x-axis are equilateral triangles.