CHAPTER 15 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:
