Volume with Washer Method: Revolving Around the x- or y-Axis

#8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

Welcome back! For this section, brush up on the basics by covering the disc method here in section 8.9 and the disk method with different axes here in section 8.10. This topic almost always appears as part of an free-response question (FRQ), so study up!

#🛁 The Washer Method

Remember the general integral format from 8.10 Volume with Disc Method: Revolving Around Other Axes? You’ll need to expand it for the washer method!

$$\int_{c}^{d}\pi (f(x)-b)^2 dx$$

So what is a washer?? Very simply, it’s a circle with another circle cut out of it in the middle.

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Picture of a blue washer with the outer radius labeled 5 cm and the inner radius labeled 3 cm.

Image courtesy of study.com

Instead of circular cross-sections to calculate an integral, we use washer cross sections when given more than one function to rotate around an axis. A washer’s area can be calculated through the equation $\pi r_1^2-\pi r_2^2$, where $r_1$ is larger than $r_2$. By replacing the circle area equation ($\pi r^2$) in the disc method with the washer area equation, we find the final washer method integral of:

$$\int_{c}^{d}\pi (f(x)-b)^2-\pi(g(x)-b)^2 dx$$

where $f(x)$ is the equation with the larger radius and $g(x)$ is the equation with the smaller radius. Remember that $f(x)-b$ is equal to the distance between the axis and function, which represents the radius.

The axis of rotation is $y = b$, the lower bound is $c$ and the upper bound is $d$. Also, on a graph, $f(x)$ should be farther from the axis of rotation than $g(x)$ in the specified interval.

Additionally, don’t forget that you need to square the functions $f(x)$ and $g(x)$ before subtracting $g(x)$ from $f(x)$. For example, imagine that a washer had an outer radius of 4 and an inner radius of 2. The total area would equal $4^2 \pi-2^...