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Use the washer method to find the volume of the solid generated when the region $R$ bounded by $y=5x$, $y=x​$, and $y=10$ is revolved about the​ $y$-axis.

I'm confused because theres no intersection besides $(0,0)$ and which is the outer and inner radius?

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    It helps to draw a picture. The radius is the distance from a point in the region to be revolved to the axis. The outer radius is the maximum distance, and the inner radius is the minimum distance. – 2017-01-27

2 Answers 2

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The region is given below.

enter image description here

In set notation, the region $R$ is given by

$$R=\{(x,y):0\leq y\leq 10 \text{ and }\frac{y}{5}\leq x\leq y\}.$$ Thus, using the Washer Method, we get

$$V=\int_0^{10}\pi\left[y^2-\left(\frac{y}{5}\right)^2\right]dy=\frac{24\pi}{25}\left[\frac{y^3}{3}\right]_{0}^{10}=\frac{24\pi}{25}\frac{1000}{3}=320\pi .$$

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The region is a triangle with vertices $(0,0), (2,10)$ and $(10,10)$. It makes $y=x$ the outer edge of each washer and $y=5x$ the inner.