When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation)
Now, when you take the volume of a 3D object, you sum the slices.
Volume = summation of all the little slices' volumes = A(x) * width
But why are you just slapping an integral as a way to do the summation? Not seeing the connection between integral being "area under the curve" and suddenly integral being "summation" of the various areas of bases.