### Home > CALC > Chapter 9 > Lesson 9.3.1 > Problem9-80

9-80.

Write the integral that calculates the volume generated when the region bounded by $y = x^2$ and $x = y^4$ is rotated:

Sketch the region and determine the points of intersection.

1. About the $x$-axis.

The outside radius $\left(R\right)$ is the curve $x = y^4$. The inside radius $\left(r\right)$ is the curve $y = x^2$. This will be a "$dx$" integral, so make sure your radii expressions are in terms of $x$.

2. About the $y$-axis.

Identify which curve will be the outside radius and which curve will be the inside radius. This will be a "$dy$" integral, so make sure your radii expressions are in terms of $y$.

3. About the line $x = 2$.

Since $x = 2$ is a vertical line, this will be similar to part (b). This time the outside radius will be $(2- R)$ and the inside radius will be $(2-r)$.