1. ## volume problems!!

1.)Consider the given curves to do the following.
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.
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.
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!

2. Originally Posted by ahawk1
1.)Consider the given curves to do the following.
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, $\pi \int_0^{10} \left( 8\right)^2 - \left( \tfrac{x^3}{125} \right)^2 dx$