Thread: volume formed by rotating the region

Find the volume formed by rotating the region enclosed by:
x=9y and y^3=x with y>=0
about the y-axis
can anyone help with this

Originally Posted by dat1611

Find the volume formed by rotating the region enclosed by:
x=9y and y^3=x with y>=0
about the y-axis
can anyone help with this

Well, the curves intersect at $\displaystyle (-3,-27), (0,0)$ and $\displaystyle (3,27)$ therefore:

By the method of rings you get

$\displaystyle V=\int\limits_0^3\pi \left[(9y)^2-(y^3)^2\right]\, dy$

By the method of cylinders you get

$\displaystyle V=\int\limits_0^{27}2\pi x(x^{1/3}-\tfrac{1}{9}x)\,dx$

Hopefully they are the same ($\displaystyle V\approx 1308.69774$).