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MULTIPLE INTEGRATION
one-eighth of the sphere. We know the formula for volume of a sphere is (4/3)πr3, so the
volume we have computed is (1/8)(4/3) π23 = (4/3) π, in agreement with our answer.
This example is much like a simple one in rectangular coordinates: the region of
interest may be described exactly by a constant range for each of the variables. As with
rectangular coordinates, we can adapt the method to deal with more complicated
regions.
We can rewrite the integral as shown because of the symmetry of the volume; this
avoids a complication during the evaluation. Proceeding:
CHAPTER 15
MULTIPLE INTEGRATION
You might have learned a formula for computing areas in polar coordinates. It is
possible to compute areas as volumes, so that you need only remember one technique.
Consider the surface z = 1, a horizontal plane. The volume under this surface and
above a region in the x-y plane is simply 1 ・ (area of the region), so computing the
volume really just computes the area of the region.
CHAPTER 15
MULTIPLE INTEGRATION
Triple Integrals
It will come as no surprise that we can also do triple integrals—integrals over a threedimensional region. The simplest application allows us to compute volumes in an
alternate
way.To approximate a volume in three dimensions, we can divide the three-dimensional
region into small rectangular boxes, each ∆x×∆y ×∆z with volume ∆x ∆y ∆z. Then we
add them all up and take the limit, to get an integral:
Of course, if the limits are constant, we are simply computing the volume of a
rectangular box.
Of course, this is more interesting and useful when the limits are not constant.
CHAPTER 15
MULTIPLE INTEGRATION