Calculus III

Instructor: Robert Vandermolen

(15.5-15.6)

Triple Integration!

The definition of triple integration is the same as the motivation for double integration, that of volume.

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If f is continuous over a bounded solid region R, the triple integral of f over R is defined as

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So for a solid region R, in 3 Dimensional space the volume of this region is given by

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=

Triple Integration!

For triple integration we have a new Fubini’s Theorem, as when f is continuous on a solid region R defined by

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where , , and are all continuous functions. Then,

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Also, we have equivalent definition for the different orders:

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dydxdz, dxdydz, dzdxdy, dxdzdy, dydzdx

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Triple Integration!

As we did with double integration, we first practice evaluating triple iterated integrals, before we move to determining bounds and manipulating such things.

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Example:

Evaluate the following:

For the first integration, hold x and y constant and integrate with respect to z.

For the second integration, hold x constant and integrate with respect to y.

Finally, integrate with respect to x.

Triple Integration!

Now, we will find it helpful later to switch the order of integration as we did with double integrals, so let’s practice finding appropriate bounds for our integration.

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Example:

Set up the triple integral for the volume of the region:

Triple Integration!

Now, we will find it helpful later to switch the order of integration as we did with double integrals, so let’s practice finding appropriate bounds for our integration.

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Example:

Set up the triple integral for the volume of the region:

First, let’s pick the order: dz dy dx

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since z has the bounds:

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we can begin by setting up the integral:

So next, we should look at what is happening in the xy-plane, to determine the remaining bounds...

Triple Integration!

Example:

Triple Integration!

Example:

Triple Integration!

Now, we will find it helpful later to switch the order of integration as we did with double integrals, so let’s practice finding appropriate bounds for our integration.

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Example:

Set up the triple integral for the volume of the region:

Now YOU TRY the order: dz dx dy

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Triple Integration!

Now, we will find it helpful later to switch the order of integration as we did with double integrals, so let’s practice finding appropriate bounds for our integration.

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Example:

Set up the triple integral for the volume of the region:

Now, let’s try a tricker order: dx dy dz

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since x has the bounds:

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we can begin by setting up the integral:

The tricky part is that now we should look at the projection (the shadow) in the yz-plane, to determine the remaining bounds...

Triple Integration!

Now, we will find it helpful later to switch the order of integration as we did with double integrals, so let’s practice finding appropriate bounds for our integration.

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Example:

Set up the triple integral for the volume of the region:

Triple Integration!

Example:

Evaluate:

The bounds for the region are

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and thus the projection onto the xy-plane is:

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Thus the bounds for the region are:

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Triple Integration!

Example:

Evaluate:

Note that after one integration in the given order, we would have to integrate

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Which is not an elementary function. To avoid this problem we can change the order of integration to dz dx dy, so that y is the outer variable...

The bounds for the region are

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and thus the projection onto the xy-plane is:

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So we have:

Now you Try!

Sketch the solid whose volume is given by the following integrals, and rewrite the integral using the indicated order of integration:

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