Application of Integration. Finding Volume: Rotating the region.

Question

I need a tutor's assistance on a homework problem. I am given y=x^2, x=y^2, and about y=1. And I am asked to find the volume by rotating the region bounded by the given curves about y=1. However, I am still unsure how to go about solving this problem. How do I solve it?

Answer 1

We have: x² ≤ y ≤ √x, 0 ≤ x ≤ 1.

By Washer Method volume V = π∫₀¹[(1 - x²)² - (1 - √x)²]dx = π∫₀¹(x⁴ - 2x²- x + 2√x)dx =

π(x⁵/5 - 2x³/3 - x²/2 + 4x^3/2/3)₀¹ = π(1/5 - 2/3 - 1/2 + 4/3) = 11π/30