Triple Integrals for Volumes of Some Classic Shapes
In the following pages, I give some worked out examples where triple integrals are used to find some
classic shapes volumes (boxes, cylinders, spheres and cones) For all of these shapes, triple integrals aren’t
needed, but I just want to show you how you could use triple integrals to find them. The methods of
cylindrical and spherical coordinates are also illustrated. I hope this helps you better understand how
to set up a triple integral. Remember that the volume of a solid region Eis given by ZZZ
E
1dV .
A Rectangular Box
A rectangular box can be described by the set of inequalities a≤x≤b,c≤y≤d,p≤z≤q. So that
the volume comes out to be length times width times height as expected:
ZZZ
E
1dV =Zb
aZd
cZq
p
1dzdydx = (b−a)(d−c)(q−p).
A Circular Cylinder
The equation for the outer edge of a circular cylinder of radius ais given by x2+y2=a2. If we want
to consider the volume inside such a cylinder with height h, then we are considering the region where
x2+y2≤a2and 0 ≤z≤h(in other words between the planes z= 0 and z=h). We already
have bounds on z, so let’s use that as the innermost integral. Now we need bounds for the circular
x2+y2≤a2in the xy-plane. We can do that in a few different ways:
1. In Cartesian Coordinates:
The solid can be described by the inequalities −a≤x≤a,−√a2−x2≤y≤√a2−x2, 0 ≤z≤h.
So we find the volume is:
ZZZ
E
1dV =Za
−aZ√a2
−x2
−√a2
−x2Zh
0
1dzdydx =Za
−a
2h√a2−x2dx = 2h1
2πa2=πa2h.
Note: I skipped some steps in the integration. You would need to see the last integration geomet-
rically (that the last integral represents the area of exactly half a circle), or you would have to use
trig substitution.
2. In Cylindrical Coordinates: A circular cylinder is perfect for cylindrical coordinates! The region
x2+y2≤a2is very easily described, so that all together the solid can be described by the
inequalities 0 ≤θ≤2π, 0 ≤r≤a, 0 ≤z≤h. So we find the volume is:
ZZZ
E
1dV =Z2π
0Za
0Zh
0
r dzdrdθ =Z2π
0
dθ Za
0
r dr Zh
0
dz = 2π1
2a2h=πa2h.
Either way, we see that we get the expected volume formula.
