Question

Find the volume of the solid of revolution generated by revolving the region bounded by y = 2x^2, y = 0, and x = 2 about the x-axis.

Answer 1

`128 \pi/5 units^3`

Explanation:

The volume of the solid revolution is expressed as;

`V = \int\limits^2_0 {\pi y^2} \, dx`

Given y = 2x²

y² = (2x²)²

y² = 4x⁴

Substitute into the formula

`V = \int\limits^2_0 {4\pi x^4} \, dx\\V =4\pi \int\limits^2_0 { x^4} \, dx\\V = 4 \pi [(x^5)/(5) ]\\`

Substituting the limits

`V = 4 \pi ([(2^5)/(5)] - [(0^5)/(5)])\\V = 4 \pi ([(32)/(5)] - 0)\\V = 128 \pi/5 units^3`

Hence the volume of the solid is
`128 \pi/5 units^3`