# Math Help - Application of integration: Volumes

1. ## Application of integration: Volumes

http://i249.photobucket.com/albums/g...ORKDIAGRAM.jpg

im supposed to find the volume generated by rotating the given region about the specified line

Find the volume generated by rotating the region between $y = x^3\text{ and }y = x^{\frac{1}{2}}$

(a) about the $y$-axis

Washers: . $V \;=\;\pi\int^1_0\left(y^{\frac{1}{3}} - y^2\right)\,dy$

Shells: . $V \;=\;2\pi\int^1_0x\left(x^{\frac{1}{2}} - x^3\right)\,dx$

(b) about $y = 1$

Washers: . $V \;=\;\pi\int^1_0\bigg[\left(1 - x^3\right)^2 - \left(1 - x^{\frac{1}{2}}\right)^2\bigg]\,dx$

Shells: . $V \;=\;2\pi\int^1_0(1-y)\left(y^{\frac{1}{3}} - y^2\right)\,dy$