solids of revolution problem

[miffman]

So I can't seem to understand why the width of anew infinitesimal cylinder is dx. Where does it come from? What does in have to do with the integral?

[Foxpup]

Because the integral of a curve is equal to the area under it, which has obvious implications when it comes to geometry. The integral ∫ 2f(x) dx is the area of the solid's cross section. The volume of the solid can be calculated by π ∫ f(x)² dx.

Consider the simplest case, a cylinder (f(x)=r). The area of the cross section (a rectangle) (∫ 2r dx) is 2r times the length, and the volume (π ∫ r² dx) is πr² times the length.

Now consider a cone whose base radius is equal to its length (f(x)=x). The area of the cross section (a triangle) (∫ 2x dx) is the square of the length, and the volume (π ∫ x² dx) is naturally πx³/3.

Does that answer your question?

[miffman]

I think I do. Thank for your response!