# Math Help - volume

1. ## volume

Find the volume formed by rotating the region enclosed by:
x=4y , y^3 = x, with y ≥ 0 about the y-axis.

thanks for any help

2. Graphing this, we can see it is symmetrical about the y-axis.

Solve each for y in terms of x:

$y=\frac{x}{4}, \;\ y=x^{\frac{1}{3}}$

Set each equal and solve for x to find the limits of integration.

Doing this, we see we get x=-8,0,8

Using shells:

$2{\pi}\int_{0}^{8}x(\frac{x}{4}-x^{\frac{1}{3}})dx$

Integrating over -8 to 0 will yield the same result.

Using washers, we can get the same thing.

${\pi}\int_{0}^{2}\left[(4y)^{2}-(y^{3})^{2}\right]dy$

3. what do you get for a final answer for volume after integrating it? my answer is off i guess

4. $\frac{512\pi}{21}$

5. thanks a lot for the help!