For simple univariate functions, integration is equivalent to "finding the area under the function".

A discrete method is a Riemann sum: Split into rectangles with height given by the integrand, sum their area:
The limit of a Riemann sum is a definite integral:
Where ranges from to . We can imagine a definite integral as a Riemann sum of infinitely thin rectangles.

This infinitely small quantity, , is called a differential.

Definite integrals typically look like this:
Fundamental theorem of calculus tells us that (assuming continuity):
Where

Sometimes the domain of integration is written at the bottom, as we saw in the rendering equation:

This notation lets us use more complex domains, such as "a hemisphere of directions"

#math #light-transport