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Mathematical Expression Editor

We compute volumes using the divergence theorem.

Work in groups of 3–4, writing your answers on a separate sheet of
paper.

So far, we’ve been using the divergence theorem to simplify the computations of
surface integrals. However, as we will see, we can also use the divergence theorem to
compute volumes of solid regions. Specifically, we will now try to compute the volume
of an ellipsoid: for and .

Give a careful sketch of the graph of :

Describe in pictures, words, interpretative dance, how the ellipsoid is drawn by as
runs from to , and runs from to .

As a gesture of friendship, I will tell you that the implicit formula for an ellipsoid is:
Use the vector-valued formula to confirm this formula.

If you were to set-up an iterated integral to compute the volume of the ellipsoid,
which coordinates would be easiest to use? Why?

Set-up an iterated integral that will compute the volume of the ellipsoid.

Now we will try to use the divergence theorem to compute volume. As a gesture of
friendship, we remind you of the statement of the divergence theorem: The first
thing we need are vector fields such that .

Find (simple!) vector fields such that .

Setting , use the divergence theorem to compute the volume of the ellipsoid.