volume problems!!

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• Jan 31st 2009, 03:11 PM
ahawk1
volume problems!!
1.)Consider the given curves to do the following. http://www.webassign.net/www14/symIm...e15fdf4024.gif
Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

2.)Consider the given curves to do the following. http://www.webassign.net/www14/symIm...7bb8fe6763.gif
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4.

3.)The region bounded by the given curves is rotated about the y-axis. http://www.webassign.net/www14/symIm...8e524ed900.gif
Find the volume V of the resulting solid by any method.

my main problem when doing these problems is i think im setting it up wrong at the beginning so can someone just help me set up these problems up to the point to the integral. Thanks in advance!
• Jan 31st 2009, 04:24 PM
ThePerfectHacker
Quote:

Originally Posted by ahawk1
1.)Consider the given curves to do the following. http://www.webassign.net/www14/symIm...e15fdf4024.gif
Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.

It will do the first one.

The intersection between $y = \tfrac{x^3}{125}$ and $y=8$ occurs when $\tfrac{x^3}{125} = 8 \implies x^3 = 125\cdot 8 \implies x = 5\cdot 2 = 10$.
Therefore, the $x$ values is going from $0\leq x\leq 10$.
Now we ought to know which curve is on top, pick a point say $x=1$. Notice that $\tfrac{(1)^3}{125} < 8$ therefore $y=8$ is on top of $y=\tfrac{x^3}{125}$.

The volume is given by, $\pi \int_0^{10} \left( 8\right)^2 - \left( \tfrac{x^3}{125} \right)^2 dx$