The region bounded by and is revolved about the axis. (a) Find the value of in the

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. x21

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. x21y

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. 21y

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. 1yx

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. 1yx2

In Exercises 16, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. yx2

In Exercises 710, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. x225

In Exercises 710, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. x225

In Exercises 710, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. 225x

In Exercises 710, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the -axis. 25x2

In Exercises 1114, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. 25x2y

In Exercises 1114, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. x2y2

In Exercises 1114, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. x2y2

In Exercises 1114, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. 2y2x

In Exercises 1518, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4. y2x3

In Exercises 1518, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4. 2x3

In Exercises 1518, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4. x34

In Exercises 1518, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4. x34x

In Exercises 1922, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. 34xy

In Exercises 1922, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. 4xy

In Exercises 1922, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. 4xyx

In Exercises 1922, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. xyx2

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. yx2

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x2x

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x2x

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 2x2

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x2

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x2x

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 2x,

In Exercises 2330, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 2x,

In Exercises 31 and 32, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x, 4

In Exercises 31 and 32, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. x, 4

In Exercises 3336, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. Verify your results using the integration capabilities of a graphing utility. , 4

In Exercises 3336, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. Verify your results using the integration capabilities of a graphing utility. 4

In Exercises 3336, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. Verify your results using the integration capabilities of a graphing utility. 40.5,

In Exercises 3336, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. Verify your results using the integration capabilities of a graphing utility. 0.5, y

In Exercises 37 40, use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 0.5, y

In Exercises 37 40, use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 0.5, yy

In Exercises 37 40, use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. 0.5, yy

In Exercises 37 40, use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. yyx

In Exercises 41 and 42, the integral represents the volume of a solid. Describe the solid. yx5

In Exercises 41 and 42, the integral represents the volume of a solid. Describe the solid. yx5

Think About It In Exercises 43 and 44, determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) x55

Think About It In Exercises 43 and 44, determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) x55x

A region bounded by the parabola and the axis is revolved about the axis. A second region bounded by the parabola and the axis is revolved about the axis. Without integrating, how do the volumes of the two solids compare? Explain. 55x

A region bounded by the parabola and the axis is revolved about the axis. A second region bounded by the parabola and the axis is revolved about the axis. Without integrating, how do the volumes of the two solids compare? Explain. 5xx

If the portion of the line lying in the first quadrant is revolved about the axis, a cone is generated. Find the volume of the cone extending from 5xx, 3

Use the disk method to verify that the volume of a right circular cone is where is the radius of the base and is the height. xx, 3

Use the disk method to verify that the volume of a sphere is x, 3

A sphere of radius is cut by a plane units above the equator. Find the volume of the solid (spherical segment) above the plane. x, 32,

A cone of height with a base of radius is cut by a plane parallel to and units above the base. Find the volume of the solid (frustum of a cone) below the plane. , 32, y

The region bounded by and is revolved about the axis. (a) Find the value of in the interval that divides the solid into two parts of equal volume. (b) Find the values of in the interval that divide the solid into three parts of equal volume. 2, y

Volume of a Fuel Tank A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of and the axis about the axis (see figure), where and are measured in meters. Find the tanks volume. 2, yy

Volume of a Lab Glass A glass container can be modeled by revolving the graph of about the axis, where and are measured in centimeters. Use a graphing utility to graph the function and find the volume of the container. 2, yy

Find the volume of the solid generated if the upper half of the ellipse is revolved about (a) the axis to form a prolate spheroid (shaped like a football), and (b) the axis to form an oblate spheroid (shaped like half of a candy). yy

Minimum Volume The arc of on the interval is revolved about the line (see figure). (a) Find the volume of the resulting solid as a function of (b) Use a graphing utility to graph the function in part (a), and use the graph to approximate the value of that minimizes the volume of the solid. (c) Use calculus to find the value of that minimizes the volume of the solid, and compare the result with the answer to part (b). y5

Water Depth in a Tank 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.) y5

Modeling Data A draftsman is asked to determine the amount of material required to produce a machine part (see figure in first column). The diameters of the part at equally spaced points are listed in the table. The measurements are listed in centimeters. (a) Use these data with Simpsons Rule to approximate the volume of the part. (b) Use the regression capabilities of a graphing utility to find a fourth-degree polynomial through the points representing the radius of the solid. Plot the data and graph the model. (c) Use a graphing utility to approximate the definite integral yielding the volume of the part. Compare the result with the answer to part (a). 5x

Think About It Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cylinder (b) Ellipsoid (c) Sphere (d) Right circular cone (e) Torus (i) (ii) (iii) (iv) (v) 5xy

Cavalieris Theorem Prove that if two solids have equal altitudes and all plane sections parallel to their bases and at equal distances from their bases have equal areas, then the solids have the same volume (see figure). 5xy

Find the volume of the solid whose base is bounded by the graphs of and with the indicated cross sections taken perpendicular to the axis. (a) Squares (b) Rectangles of height 1 xy

Find the volume of the solid whose base is bounded by the circle with the indicated cross sections taken perpendicular to the axis. (a) Squares (b) Equilateral triangles (c) Semicircles (d) Isosceles right triangles xyx

The base of a solid is bounded by and Find the volume of the solid for each of the following cross sections (taken perpendicular to the axis): (a) squares, (b) semicircles, (c) equilateral triangles, and (d) semiellipses whose heights are twice the lengths of their bases. yx

Find the volume of the solid of intersection (the solid common to both) of the two right circular cylinders of radius whose axes meet at right angles (see figure). Two intersecting cylinders Solid of intersection x

A manufacturer drills a hole through the center of a metal sphere of radius The hole has a radius Find the volume of the resulting ring. x

For the metal sphere in Exercise 65, let What value of will produce a ring whose volume is exactly half the volume of the sphere? xx

In Exercises 6774, find the volume generated by rotating the given region about the specified line. x

In Exercises 6774, find the volume generated by rotating the given region about the specified line. x10

In Exercises 6774, find the volume generated by rotating the given region about the specified line. x10

In Exercises 6774, find the volume generated by rotating the given region about the specified line. x10y

In Exercises 6774, find the volume generated by rotating the given region about the specified line. 10y

In Exercises 6774, find the volume generated by rotating the given region about the specified line. 10yx

In Exercises 6774, find the volume generated by rotating the given region about the specified line. yx3

In Exercises 6774, find the volume generated by rotating the given region about the specified line. yx3

The solid shown in the figure has cross sections bounded by the graph of where (a) Describe the cross section when and (b) Describe a procedure for approximating the volume of the solid. x33

Two planes cut a right circular cylinder to form a wedge. One plane is perpendicular to the axis of the cylinder and the second makes an angle of degrees with the first (see figure). (a) Find the volume of the wedge if (b) Find the volume of the wedge for an arbitrary angle Assuming that the cylinder has sufficient length, how does the volume of the wedge change as increases from to x33x

(a) Show that the volume of the torus shown is given by the integral where (b) Find the volume of the torus. 33x2