Question

What is the washer method formula?

Answer 1

In this problem, we have to find what is the washer method formula. Here we should first know about disc integration, in which the washer method is a part. We should know that the disc integration models the resulting three-dimensional shape as a stack of an infinite number of discs. It is possible to use the same principle with rings instead of disc, which is the washer method to obtain hollow solids of revolution. Here we can see about the definitions and the formulas.

Complete step-by-step solution:
We can write the formula for disc method which is used to find the volume of solid of revolution with the function of x
$\pi \int\limits_{a}^{b}{R{{\left( x \right)}^{2}}dx}$
Where $R\left( x \right)$ is the distance between the function and the axis of rotation, which works only if the axis of rotation is horizontal (if the function is y, then the axis of rotation is vertical).
We can now write the formula for the washer method which is used to obtain hollow solids of revolution.
It is the subtraction of the volume of the inner solid of revolution from the outer solid of revolution, which can be calculated in a single integral.
$\pi \int\limits_{a}^{b}{{{R}_{O}}{{\left( x \right)}^{2}}-{{R}_{I}}{{\left( x \right)}^{2}}dx}$
Where ${{R}_{O}}\left( x \right)$ is the function that is farthest from the axis of rotation and ${{R}_{I}}\left( x \right)$ is nearest from the axis of rotation.

Note: We should know that the disc integration models the resulting three-dimensional shape as a stack of an infinite number of discs. It is possible to use the same principle with rings instead of disc which is the washer method to obtain hollow solids of revolution.