Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

x=2-y², x=y⁴; about the y-axis

Answer 1

Intersection Points of the Two Curves

The curves x = 2 – y² and x = y⁴ intersect when 2 – y² = y⁴

which means that y⁴ + y² - 2 = (y² – 1)(y² + 2) = 0 where

the only real solutions are y = ±1

So the curves x = 2 – y² and x = y⁴ intersect at points (1,–1) and (1, 1)

Cross-Sectional Area Function

A(y) = π[(2 – y²)² – (y⁴)²] = π[4 – 4y² + y⁴– y⁸]

Volume