A solid can be thought of as a "stack" of many, many cross sections with area A and thickness dx.
The area A of a cross section is often changing though, so you have to use integration to get the total volume of the solid.
That is, V = ∫ A(x) dx.
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The challenge here is often finding your function A(x). This will be different depending on whether you are using square, triangular, or circular cross sections in the problem.
You need to express A(x) as a function of your function on the xy plane, f(x).
If you are using square cross sections, then you will substitute in [f(x)]2 since the area of a square is s2.
For triangular cross sections, substitute in 1/2 * [f(x)]2 since A = 1/2 * b * h.
For circular cross sections, this happens to be equal to the disk (or washer) method!
Hope this kind of builds some intuition.