volume formed by rotating the region

**dat1611** · August 7th 2009, 07:28 AM

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

**Failure** · August 7th 2009, 07:53 AM

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 $(-3,-27), (0,0)$ and $(3,27)$ therefore:

By the method of rings you get

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

By the method of cylinders you get

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

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

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