Confusion with integration?

so, we integrate the area function A(x) of a solid [say a pyramid] that is integrable over the distance 'x' [say from 0 to 3] we get its volume. my doubt is the cross section of the solid is a square here lying in the yz pane. so the pyramid points to the x axis. now why did they take the area as a function of x? instead of yz. also is this method possible only for linear increasing solids so that a cone is OK but a praboloid is not??????? confUSinG