Question

Find the volume of the solid obtained by rotating region bounded by the given curves about the line. y = x, y = 0, x = 2, x = 5; about x = 1?

Answer 1

The volume is given by the integral

`\displaystyle2\pi\int_2^5x(x-1)\,\mathrm dx`

using the shell method. We approximate the solid of revolution with hollow cylinders with radius
`x-1` and height
`x`.

`\displaystyle2\pi\int_2^5x^2-x\,\mathrm dx=2\pi\left(\frac{x^3}3-\frac{x^2}2\right)\bigg|_(x=2)^(x=5)=57\pi`