Integration of Scalar Fields

Instead of finding the area under a curve, we seek the volume under a scalar field.

It's the same idea as single variable integration (Riemann Sum)

Definition

Notation

It's common to see the above written as:

Fact

Since .

How do we do this in practice?

Example

Find the volume under the scalar field of

solution

Note

It can be shown that if we go the other way (i.e integrate along x first, then y), we would get the same answer (which makes sense geometrically too).

The domain does not have to be a rectangle:

Generally:

Visualization

Example

Evaluate with R: area constrained between and

solution
Recall Area Between Two Curves. Our domain (constraints) looks like this:

Example

Evaluate with R: area constrained between and :

solution

Example

Evaluate :

solution
It's impossible to do it in this order, but we can switch the bounds to make it possible:

Our domain:

Let . Our bounds stay the same, since and

(see half angle identities)