Volumes by slices and volume of cylinder

Question

So I am looking at this method of finding the volume of a solid by slices. The idea is to create slices/slabs which are cylinders/prisms and integrate them.

My problem is that in the book I'm using the volume of a cylinder/prims/slab is assumed to be: $$V=Ah\tag{1}$$ so the volume is then: $$\int_a^bA(x)dx$$

This method is then used to show that the volume of a cylinder is equation $(1)$. That is circular reasoning.

Am I missing something obvious or is the volume of a cylinder proved with more rigorous math to follow?

Answer 1

Here is the reasoning:

1. Define the volume of the prism to be $A(x)h$. This definition agrees with our intuition of volume.
2. We want the integral to compute volumes in a more abstract way. However, we want the integral to still give us our intuitive results like the volume of the prism is $A(x)h$, where $A(x)$ is the area of the base.
3. We compute the integral and huzzah! We get our original result!

In this way, what you are doing is more like verifying that the tools you've made do what you want them to do (compute volumes, and have these volumes agree in the selected ``intuitive" cases).

Of course, you could start by defining the area of a prism to be $\int_a^bA(x)\;\text{d}x$, and then as a theorem you get that the volume is $A(x)h$.

All depends on what you want to be theorems and definitions.