I am trying to solve a question in my integration topic using volumes of solids of revolutions the attatchment shows my workings and the part I would like some guidance with
For part 2,
$\displaystyle 0.5\int_{y=0}^{54}{\pi}x^2dy=\frac{{\pi}}{2}\int_{ 0}^{54}\left(\frac{y}{2}\right)^{\frac{2}{3}}dy$
integrating this, we bring out the denominator under y first, then raise the power of y by 1
and divide by the new power
$\displaystyle \frac{{\pi}}{2}\int_{0}^{54}\left(\frac{1}{2}\righ t)^{\frac{2}{3}}y^{\frac{2}{3}}dy=\frac{{\pi}}{2}\ left(\frac{1}{4}\right)^{\frac{1}{3}}y^{\frac{2}{3 }+\frac{3}{3}}\frac{1}{\frac{2}{3}+\frac{3}{3}}$
evaluated from y=0 to y=54
$\displaystyle \frac{{\pi}}{2}\left(\frac{1}{4}\right)^{\frac{1}{ 3}}y^{\frac{5}{3}}\left(\frac{3}{5}\right)$
from 0 to 54
$\displaystyle =\frac{3{\pi}}{10}\left(\frac{1}{4}\right)^{\frac{ 1}{3}}54^{\frac{3}{5}}-0$