# Thread: 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

a) R1 about OC
b) R1 about BC

2. Hello, sobadin!

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

(a) about the $\displaystyle y$-axis

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

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

(b) about $\displaystyle y = 1$

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

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