While I'm at it....

C'mon guys... you don't have to pay attention to my posts about math. You can post about your own topics. I'd be glad to give input on your topics of interest. While you decide what you're going to post about, I'm going to go into the current calculus work.

Rotational Volume. I'm not sure what you use it for, but I know that it has applications somewhere, mainly with torque and other spinny things.

Rotational volume is the volume that some 2 dimensional object covers as it rotates completely around an axis.

Any rotational volume can be represented as a Mean Value Cylinder if the 2 dimensional object is rotating around the x axis. A mean value cylinder has the same volume as the rotational volume, but it's volume can be calculated by using geometry. The radius of the mean value cylinder is the average value of the function over the given interval and the height of the cylinder is the length of the interval.

An illustration should help you out.

Here, we see some function integrated, then the mean value is found and a rectangle is created. The shape that a rectangle forms when rotated about an axis is a cylinder, which appears on the right. The rotational area can then be calculated by squaring the mean value and multiplying it by B-A and by π. This is because the volume of a cylinder is V= πr²h

The mean value is represented as the following equation:

1/(B-A) * ∫f(x) dx

The height is B-A

Therefore, the formula for rotational volume is:

V = π*(1/(B-A) * ∫f(x) dx)²*(B-A)

or

V = π/(B-A) * (∫f(x) dx)²

As for areas that are inclosed by two integrals, its basically the same thing, except that one of the cylinders is positive and one is negative. You add their volumes together and the result represents the rotational volume of some area between two curves.