# Different ways of calculating a solid of revolution

1. Oct 7, 2012

### microberry1

In my calculus exam, I would like to know for all of the questions that I am 100% right (who wouldn't?). I can be sure of this for most basic questions using my graphics calculator (fx-9750gii) and some simple maths. One thing I want to know how to be "completely sure" to solve are solid of revolution questions. Unlike basic area integrals, I can't do them in the calculator, so I want to find another way.

I thought that one way might be to find the average value of the function over the closed interval to be rotated, then use the volume of a cylinder formula (V=(1/3)∏r^2h) and substituting (width of closed interval)=h and (average value of f(x))=r. Trying this out with some basic problems, this did not work, e.g. ∏∫[between 0 & 4] ((x^2)^2) dx ≠ 4∏(16/3)^2 (RHS is using the method mentioned). My first question; why does this method not work?

Also, are there any other methods to calculate the solid of revolution in an exam other than the usual formula V=∏∫[between a & b](f(x))^2 dx, that I can use to double-check my answers?

I hope the post is clear enough, sorry for not knowing how to express the integrals better...

Thanks
microberry1