Volumes of revolution: y = x^2, x = y^2, about x = 3

[johnboy]

I thought I knew how to do volume problems, but I messed up my last homework on them pretty badly. For cross sections and shells, are there any general forms using constants? I don't know if that makes sense, but I am having trouble determining how to do volumes revolved about lines other than the x- and y-axis. For example:

. . .y = x^2
. . .x = y^2
. . .about x = 3

I know if it's not revoled about anything, then it would just be pi integral (y^0.5)^2 -(y^2)^2 dy. If anyone has any suggestions on the lines revolved about x or y= c, I would greatly appreciate it. Thanks :-/

 

[galactus]

A good thing to remember about shells is the cross sections are parallel to the axis about which it is revolved.

If you're revolving about x=3, a vertical line, then the cross sections are parallel to x=3 and vertical also...stacked up along the x-axis.

In other words, if revolving about an axis other than the x or y axis, the integral is with respect to the same line about which you're revolving.

For example, if you revolve about x=3, then the integral should be in terms of x. If about y=3, then in terms of y.

I know it can be daunting to picture what's going on, but with a little practice you get better at it.

Shells:

\(\displaystyle \L\\2{\pi}\int_{0}^{1}\underbrace{(x-3)}\underbrace{(x^{2}-sqrt{x})}\underbrace{dx}\)
\(\displaystyle \ \;\ \;\ \text{radius\;\ height\;\ thickness}\)

Washers:

\(\displaystyle \L\\{\pi}\int_{0}^{1}((3-y^{2})^{2}-(3-\sqrt{y})^{2})dy\)

 

[johnboy]

if we revolved that same equation about x=-5

would it be pi S (5+y^.5)^2-(5+y^2)^2 ? Thanks again..

 

[galactus]

Yep.