Question: An area in the first quadrant is bounded by the ellipse $\displaystyle 4x^2+9y^2=36 $ and the axes. This area is rotated through four right angles about the x-axis. Find

(a) the volume of the solid generated

(b) the x-coordinate of the centre of gravity of this solid

My attempt:

a) $\displaystyle 4x^2+9y^2=36 $

$\displaystyle \frac{x^2}{9}+ \frac{y^2}{4 }=1 $

$\displaystyle y =\frac{\sqrt{4-\frac{4}{9}x^2}}{2} $

element of volume = $\displaystyle \pi y^2 \partial x $

v =$\displaystyle \int_{0}^{3} \pi \left ( \frac{\sqrt{4-\frac{4}{9}x^2}}{2} \right )^2 \mathrm{d} x $

v =$\displaystyle 2 $

b) $\displaystyle \bar{x}\int_{0}^{3} \pi \left ( \frac{\sqrt{4-\frac{4}{9}x^2}}{2} \right )^2 \mathrm{d} x = \int_{0}^{3} \pi x \left ( \frac{\sqrt{4-\frac{4}{9}x^2}}{2} \right )^2 \mathrm{d} x $

$\displaystyle \bar{x}\int_{0}^{3} \left ( \frac{{4-\frac{4}{9}x^2}}{4} \right ) \mathrm{d} x = \int_{0}^{3} x - \frac{x^3}{9} \mathrm{d} x $

$\displaystyle \bar{x} = \frac{9}{8} $

but the book gives the answer for a) $\displaystyle 8 \pi $ b) $\displaystyle \frac{4}{5} $

please can i have some help with this question