Volumes of Revolution (College Board AP® Calculus AB): Exam Questions

Let be the region enclosed by the graphs of and , the -axis, and the vertical line , as shown in the figure above.

Write, but do not evaluate, an integral expression that gives the volume of the solid generated when is rotated about the horizontal line .

Let be the region in the first quadrant enclosed by the coordinate axes and the graph of , as shown in the figure above.

Find the volume of the solid generated when is rotated about the -axis.

The inside of a vase of height inches has circular cross sections, as shown in the figure above. At height , the radius of the vase is given by , where . The units of and are in inches.

Find the volume of the vase.

Let be the region in the first and second quadrants bounded above by the curve of and below by the horizontal line .

Find the volume of the solid generated when is rotated about the -axis.

Let be the region in the first quadrant enclosed by the graphs and on the interval as shown in the figure above.

Write, but do not evaluate, an integral expression for the volume of the solid generated when is rotated about the vertical line .

A company designs fishing lures using the family of curves with equation , where is a positive constant. The figure above shows the region in the first quadrant bounded by the -axis and the curve , for some . Each fishing lure is in the shape of the solid generated when such a region is revolved about the -axis. Both and are measured in inches.

For a particular fishing lure, the volume is cubic inches. What is the value of for this fishing lure?

Let be the region in the first quadrant enclosed by the -axis, the line , and the graphs of and , as shown in the figure above.

Find the volume of the solid generated when is rotated about the -axis.

Let be the region in the first quadrant enclosed by the coordinate axes, the line , and the graphs of and , as shown in the figure above.

Find the volume of the solid generated when is rotated about the -axis.

Let and be the region in the fourth quadrant bounded below by the graph , above by the -axis and to the left by the -axis.

Find the volume of the solid generated when is rotated about the horizontal line .

Let and be the functions given by and . Let be the shaded region in the first quadrant enclosed by the -axis and the graphs of and , and let be the shaded region in the first quadrant enclosed by the graphs of and as shown in the figure above.

Find the volume of the solid generated when and are revolved around the horizontal line .