# What is the washer method formula?

Jul 30, 2015

The general "formula" is integrate (area of large disk - area of small disk) times thickness.

#### Explanation:

Any attempt to give a more specific "formula" looks very complicated to me.

Here are a couple of pointers that may be helpful:

For the Washer Method

The area of a disk is the area of a circle. The volume is the area $\pi {r}^{2}$ times the thickness, which will be either $\mathrm{dx}$ or $\mathrm{dy}$ depending on the problem. So it will be either $\pi {r}^{2} \mathrm{dx}$ or $\pi {r}^{2} \mathrm{dy}$

So, for some $a$ and $b$, we'll have ${\int}_{a}^{b} \left(\pi {R}^{2} - \pi {r}^{2}\right) d \text{x or dy}$
Where $R$ is the radius of the larger disk and $r$ that of the smaller.

The functions used for the radii, must have the same independent variable as the differential.

If revolving around a horizontal line $y = k$, the thickness of the representative disk will be the differential $\mathrm{dx}$.
In this case:
The larger disk will be determined by the function $y = g \left(x\right)$ farther from the line $y = k$ and the smaller disk, by the function $y = f \left(x\right)$ closer to $y = k$

If revolving around a vertical line $x = h$, the thickness of the representative disk will be the differential $\mathrm{dy}$.
In this case:
The larger disk will be determined by the function $x = g \left(y\right)$ farther from the line $x = h$ and the smaller disk, by the function $x = f \left(y\right)$ closer to $x = h$

If the graphs of the functions cross each other of the line about which we are rotating, that must be taken into account.

If two region must be broken into two or more pieces to, then the volume must be calculated by using two or more integral.