You're familiar with using a definite integral, $\int_a^b f(x) \,dx$, to find the area under a curve. This works by summing up an infinite number of infinitesimally thin rectangles over an interval on the x-axis.

Now, let's extend this idea to three dimensions. Instead of a function $f(x)$ defining a curve, we have a function $f(x,y)$ that defines a surface, like a sheet draped over the xy-plane. Our goal is no longer to find the area under a curve, but the volume of the solid region between this surface and a defined area on the xy-plane.