Integration of Scalar Fields

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

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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

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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:

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Generally:

Visualization

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Example

Evaluate with R: area constrained between and

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

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Example

Evaluate with R: area constrained between and :

solution

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Example

Evaluate :

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

Our domain:

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Let . Our bounds stay the same, since and

(see half angle identities)