Volume of a Pyramid

1) Area Moving Through Space

Below is a diagram of a cone made up of a polygon in the $xy$-plane shrunk to a point on the $z$-axis.

As the polygon rises, it shrinks proportionally to its closeness to the top point.

2) Approximating with Riemann Sums

Riemann sums are a nice way to think about integrals. They approximate area under the curve, and regular integrals are just limits of these sums as $n$, the number of subdivisions, approaches $\infty$.

Riemann Sum Review: To estimate the area under the curve, we chop up our interval into $n$ equal pieces, pick sample points from each of those pieces, and add up the areas of those rectangles.

3D Riemann Summs:

Because we already know how to find the volume of a prism, we can use this to approximate the volume of an object like this with Riemann sums. You can think of these prisms are the higher-dimensional analogue of the bars you normally compute the area of while computing a Riemann integral.