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Mathematical Expression Editor

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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.

Triangles

Let be the triangle with vertices and and . Then we can write as but we could
also write Which of these might be a better choice to compute

Based on
these two different descriptions of , we can evaluate the double integral
through two rather different looking iterated integrals. Write with me,
If we were to use the first description of , we might have trouble finding
an -antiderivative of , so let’s try the second description. In that case,

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

Evaluate the integral where is the quarter circle,

We use rectangularpolar coordinates.

Again consider the region How does compare to

The region is symmetric across the line . As a consequence of this, we might have
computed that and likewise . Because of this, and the fact that the integral of a sum
is the sum of integrals, we could have deduced

Spheres and hemispheres

Let be the region .

Explain why .

This integral vanishes because integral over the northern hemisphere of will cancel
the contribution from the southern hemisphere of .

Let be the region

Show that .

Unlike the previous example, this does not vanish.

We use

cylindricalspherical coordinates.

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

Indeed, because, for points , we have and . Moreover, except when , it is the case
that . By comparing the integrands, we can gain insight into the relative sizes of the
integrals.

This same kind of thinking can lend insight into the question of what happens when
when is very large.