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

Answer:

$128 \pi/5 units^3$

Step-by-step 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

$V = \int\limits^2_0 {4\pi x^4} \, dx\\V =4\pi \int\limits^2_0 { x^4} \, dx\\V = 4 \pi [\frac{x^5}{5} ]$

Substituting the limits

$V = 4 \pi ([\frac{2^5}{5}] - [\frac{0^5}{5}])\\V = 4 \pi ([\frac{32}{5}] - 0)\\V = 128 \pi/5 units^3$

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