aphethwevdz

2023-03-11

What is the washer method formula?

Karbamidjts

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 dx or dy depending on the problem. Thus, it will be either $\pi {r}^{2}dx$ or $\pi {r}^{2}dy$
Thus, 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 radii functions 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 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 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 $x=h$
It must be taken into account if the graphs of the functions cross each other of the line about which we are rotating.
If a region must be divided into two or more parts, the volume must be calculated using two or more integrals.

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