Finding Volumes

[Guest]

I need help with this problem for my calc class. Any help would be greatly appreciated! Thank you!

The problem is "A solid is formed by rotating around the y-axis the region that is bounded by the graphs of y=x^4 and y=8x. Find the exact volume of the solid using the fundamental theorem of calculus. Assume that x and y are in inches."

 

[galactus]

Are you familiar with volumes by cylindrical shells?. That may be a way to

go.

Setting the two equations equal to one another and solving for x:

\(\displaystyle x^{4}=8x\), \(\displaystyle x^{3}=8\), therefore x=0 or 2.

At each x in [0,2] the cross-section of the region between x^4 and 8x is

parallel to the y-axis. It produces a cylindrical surface of height 8x-x^4

and radius x. The area of this surface is \(\displaystyle 2{\pi}x(8x-x^{4})\).

The volume of the solid is:

\(\displaystyle \L\\2{\pi}\int_{0}^{2}x(8x-x^{4})dx=2{\pi}(F(2)-F(0))\)

Perform the integration.

Aa a check, do it the other way, by slices, with respect to y:

\(\displaystyle \L\\{\pi}\int_{0}^{16}((y^{\frac{1}{4}})^{2}-(\frac{y}{8})^{2})dy={\pi}(F(16)-F(0))\)

Integrate and you should get the same thing.

I suppose this is what they're meaning when they say use the FFTC