Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be accurately determined using the concept of solids of revolution. For example, consider a curved function in a two-dimensional plane. Rotating this curve about the x-axis generates a symmetrical three-dimensional shape known as a solid of revolution. The line around which the region revolves is the axis of revolution.

When the region lies directly against the axis, it produces solid circular cross-sections. To determine the volume of such a solid, the disk method may be used.

This method divides the solid into thin slices perpendicular to the axis. Each slice becomes a disk with a small width and radius equal to the distance from the axis to the curve.

The volume of each disk is obtained by multiplying its area by its thickness. The total volume of the solid is approximated by summing the volumes of all the circular disks formed.

As the number of disks increases and their thickness approaches zero, the summation transitions into the definite integral of the cross-sectional area over the given interval, yielding the exact volume.

The variable of integration and the expression for radius depend on the orientation of the axis.