Question

Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified lines. y = 3 root x, y = 3x A) About the y-axisB) About the line y=3

Answer 1

See below for answers

Explanation:

Part A

`\displaystyle V=\pi\int\limits^d_c {(R^2-r^2)} \, dy\\=\pi\int\limits^3_0 {\biggr(\biggr((y^2)/(9)\biggr)^2-\biggr((y)/(3)\biggr)^2\biggr)} \, dy\\=\pi\int\limits^b_a {\biggr((y^4)/(81)-(y^2)/(9)\biggr)} \, dy`

Make sure to write the functions in terms of y since you rotate on a vertical axis.

Part B

`\displaystyle V=\pi\int\limits^b_a {(R^2-r^2)} \, dx\\=\pi\int\limits^1_0 {((3-3√(x))^2-(3-3x)^2)} \, dx\\=\pi\int\limits^1_0 {((9-18√(x)+9x)-(9-18x+9x^2))} \, dx\\=\pi\int\limits^1_0 {(-18√(x)-9x-9x^2)} \, dx`

Make sure to fix the radii so they start at y=3.