Volumes of Revolution - Ellipsoid

[Noir]

Homework Statement

An ellipse is rotated around the y-axis, find the volume of this solid.

Homework Equations

x^2 / a^2 + y^2 / b^2 = 1
$$\pi\int_{a}^-a x^2 dy$$

The Attempt at a Solution

I'm having trouble solving this; I know that the upper and lower bounds of the curve occur on the y-axis so I think that the ellipse equation can be rearranged to form;
x^2 / a^2 + 0 = 1
Thats where the upper and lower bounds for the integral come from.
However I know that the volume of an ellipsoid is V = 4/3*pi*a*b^2.
I just can't seem to get it to work.

Any help is appreciated :)

 

[rock.freak667]

Shouldn't your limits be ±b ?

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \Rightarrow \frac{x^2}{a^2}=1-\frac{y^2}{b^2}$$

find x² in terms of y², then put that into π∫x² dy from b to -b and calculate.