Disc integration

This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness.

It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions.

This can be calculated in a single integral similar to the following: where RO(x) is the function that is farthest from the axis of rotation and RI(x) is the function that is closest to the axis of rotation.

For example, the next figure shows the rotation along the x-axis of the red "leaf" enclosed between the square-root and quadratic curves: The volume of this solid is: One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

One simply must solve each equation for x before one inserts them into the integration formula.