
Volume of a Solid
I was hoping someone could doublecheck this for me:
Find the volume of a solid obtained by rotating the region bounded by y = x^2 + 1 , y = 9  x ^2, about y = 1.
(Sorry I don't know how to use the symbols, so this may be hard to read)
pi integral from 5 to 5 ( ( 1+ 9x^2)^2  ( 1 + x^2 + 1 )^2) dx
After some algebra...
pi (96 x  (4/3) x ^3) from 5 to 5
626.66 pi
Not sure if I am doing this correctly, would appreciate some feedback.

recheck your limits of integration ...
$\displaystyle x^2 + 1 = 9  x^2$
$\displaystyle 2x^2  8 = 0$
$\displaystyle 2(x+2)(x2) = 0$
using symmetry ...
$\displaystyle V = 2\pi \int_0^2 [(9  x^2) + 1]^2  [(x^2 + 1) + 1]^2 \, dx = 256\pi
$

I see what I did wrong. Thsnks for your help.
There's one more thing I need help with...
The base of a solid is a square with vertices (1,0), (0,1), (0, 1), and (1, 0). Each crosssection perpendicular to the xaxis is a semicircle. Find the volume.
I was attempting to find a quarter of the volume and just multiply by 4. I found the line above the xaxis to be y = 1x, so attempting to use that, i came up with
8pi * Integral (1x^2)(1x) dx

I would do half the volume and double.
on the positive side of x ...
region is bounded above by y = 1x and below by y = x1
diameter = (1  x)  (x  1) = 2(1  x)
radius = (1  x)
crosssectional area = $\displaystyle \frac{\pi}{2}(1x)^2$
$\displaystyle V = 2 \int_0^1 \frac{\pi}{2}(1x)^2 \, dx$
you should be able to finish from here.

OK, this wasn't finished after all...
On the first one:
Using the new limits of integration, I come up with 362.666 pi, not 256. Here are my last few steps:
2pi Integral from 0 to 2 96 4x^2
Integrated to 96x (4x^3)/3 from 0 to 2
19210.66 * 2pi
Why do we have different answers?

$\displaystyle V = 2\pi \int_0^2 [(9  x^2) + 1]^2  [(x^2 + 1) + 1]^2 \, dx $
$\displaystyle V = 2\pi \int_0^2 (10  x^2)^2  (x^2 + 2)^2 \, dx$
$\displaystyle V = 2\pi \int_0^2 [(100  20x^2 + x^4)  (x^4 + 4x^2 + 4)^2 \, dx$
$\displaystyle V = 2\pi \int_0^2 (96  24x^2) \, dx$
$\displaystyle 2\pi\left[96x  8x^3\right]_0^2 = 2\pi(192  64) = 2\pi(128) = 256\pi$

I just dropped a 0 for no apparent reason. Too much time studying. Thanks again!