y-First iterated integral over a rectangle
Suppose we wish to compute over a rectangle . We could first compute , holding constant. Then, computing gives us the answer. In other words,
which we just write as .
This is called an iterated integral. Fubini's Theorem tells us that if is continuous throughout , then
.
That is, the order of integration does not matter.
In this interactive figure we demonstrate an iterated integral by first integrating with respect to then integrating with respect to . Click through each of the checkboxes in order and see why the iterated integral is a valid way of computing the double integral.
Developed for use with Thomas' Calculus, published by Pearson.