552 CHAPTER 27. THE INTEGRAL IN OTHER COORDINATES
12. Compute the volume of a sphere of radius R using cylindrical coordinates.
13. Fill in all details for the following argument that∫∞
0e−x2
dx =12√
π.
Let I =∫
∞
0 e−x2dx. Then
I2 =∫
∞
0
∫∞
0e−(x2+y2)dxdy =
∫π/2
0
∫∞
0re−r2
dr dθ =14
π
from which the result follows.
14. Show that∫
∞
−∞
1√2πσ
e−(x−µ)2
2σ2 dx = 1. Here σ is a positive number called the standarddeviation and µ is a number called the mean.
15. Show using Problem 13 that Γ( 1
2
)=√
π . Recall Γ(α)≡∫
∞
0 e−ttα−1dt.
16. Let p,q > 0 and define B(p,q) =∫ 1
0 xp−1 (1− x)q−1. Show that
Γ(p)Γ(q) = B(p,q)Γ(p+q) .
Hint: It is fairly routine if you start with the left side and proceed to change variables.
27.7 The Moment of Inertia and Center of MassThe methods used to evaluate multiple integrals make possible the determination of centersof mass and moments of inertia for solids. This leads to the following definition.
Definition 27.7.1 Let a solid occupy a region R such that its density is ρ (x) for xa point in R and let L be a line. For x ∈ R, let l (x) be the distance from the point x to theline L. The moment of inertia of the solid is defined as
I =∫
Rl (x)2
ρ (x)dV.
Letting (x,y,z) denote the Cartesian coordinates of the center of mass,
x=
∫R xρ (x)dV∫R ρ (x)dV
, y =
∫R yρ (x)dV∫R ρ (x)dV
, z =
∫R zρ (x)dV∫R ρ (x)dV
where x,y,z are the Cartesian coordinates of the point at x.
The reason the moment of inertia is of interest has to do with the total kinetic energyof a solid occupying the region R which is rotating about the line L. Suppose its angularvelocity is ω . Then the kinetic energy of an infinitesimal chunk of volume located at pointx is 1
2 ρ (x)(l (x)ω)2 dV . Then using an integral to add these up, it follows the total kineticenergy is
12
∫R
ρ (x) l (x)2 dV ω2 =
12
Iω2
Thus in the consideration of a rotating body, the moment of inertia takes the place of masswhen angular velocity takes the place of speed.
As to the center of mass, its significance is that it gives the point at which the mass willbalance. To see this presented in terms of point masses, see Definition 14.5.4. Here thesums are replaced with integrals.