We practice more computations and think about what integrals mean.

In this section we will continue to set-up (and sometimes compute) double and triple integrals and think about what these mean.


It’s important to do a self-check to see if our purported value for an integral is at all plausible.

The region is a triangle with base and height , so the area of the region is which is about square units. In other words, which also means that We are claiming that equals , which is about .

When , the value of is sometimes positive, sometimes negative, but at least we know that and this inequality then implies that So is certainly in the ballpark of plausibility.

Polar coordinates

Again consider the region How does compare to

Spheres and hemispheres

Again let be the region Set and . How does relate to ?