Question

Let Ohm be the region bounded by the function y = -x^2 + 2x and the x-axis. This region is revolved about the y-axis. Find the volume of the solid formed.

a. 4 pi/3
b. 8 pi/3
c. 11 pi/3
d. 512 pi/15
e. 16 pi/15
f. None

Answer 1

Option b

Explanation:

The intersection of the given curve with x axis are

when y =0

i.e. when

`-x^2 + 2x=0\\x=0,2`

Since rotated about y axis we use method of cylinders

Volume =
`\int\limits^2_0 {2\pi xy} \, dx \\=2\pi \int\limits^2_0 x(-x^2+2x)dx\\= 2\pi((-x^3)/(3) +x^2)\\= (8\pi)/(3)`

option b is right