Practice Volumes of Revolution in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.
Spin a flat region around a line, like spinning a pottery wheel. The flat shape sweeps out a 3D solid. To find its volume, slice the solid into thin pieces (discs, washers, or shells), find the volume of each slice, and add them up—which means integrate.
Example 1
easyFind the volume of the solid formed by rotating f(x) = \sqrt{x} around the x-axis from x=0 to x=4.
Example 2
hardFind the volume when the region between y=x and y=x^2 (0 \leq x \leq 1) is rotated around the x-axis.
Example 3
easyFind the volume of the solid formed by rotating y = 2x from x=0 to x=3 around the x-axis.
Example 4
mediumUse the shell method to find the volume when the region bounded by y = x^2, x=0, y=1 is rotated around the y-axis.