Different ways of calculating a solid of revolution

[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

 

[lippyka]

Regarding your first question, the method that you proposed does not work because it does not take into account the changing radius of the solid of revolution. The volume of a cylinder formula assumes that the radius of the cylinder is constant, but when rotating a function about an axis, the radius changes as you move along the interval. Therefore, the method that you proposed does not work to calculate the volume of a solid of revolution. Regarding your second question, there are several other methods that you can use to calculate the solid of revolution in an exam. One such method is called the disc method, which involves dividing the interval into many small sections, and using the formula for the volume of a thin disc (V = πr^2h) to calculate the volume of each individual disc. Summing up all of the disc volumes gives you the total volume of the solid of revolution. Another method is the washer method, which involves subtracting the volume of a smaller "inner" solid of revolution from the volume of a larger "outer" solid of revolution. This method works especially well for calculating the volume of a solid of revolution with inner and outer boundaries.