Mathematical Tools for Physics

8—Multivariable Calculus 223

A Moment of Inertia
The moment of inertia about an axis is

∫

r⊥^2 dm. What is the moment of inertia of a uniform sheet of massM
in the shape of a right triangle of sidesaandb? Take the moment about the right angled vertex. The area mass
density,σ=dm/dAis 2 M/ab. The moment of inertia is then

∫

(x^2 +y^2 )σ dA=

∫a

0

dx

∫b(a−x)/a

0

dy σ(x^2 +y^2 ) =

∫a

0

dxσ

[

x^2 y+y^3 / 3

]b(a−x)/a

0

=

∫a

0

dxσ

[

x^2

b

a

(a−x) +

1

3

(

b

a

) 3

(a−x)^3

]

=σ

[

b

a

(

a^4

3

−

a^4

4

)

+

1

3

(

b^3

a^3

a^4

4

)]

=

1

12

σ

(

ba^3 +ab^3

)

=

M

6

(

a^2 +b^2

)

The dimensions are correct. For another check take the case wherea= 0, reducing this toMb^2 / 6. But wait, this
now looks like a thin rod, and I remember that the moment of inertia of a thin rod about its end isMb^2 / 3. What
went wrong? Nothing. Look again more closely. Show why this limiting answer ought to be less thanMb^2 / 3.

Volume of a Sphere
What is the volume of a sphere of radiusR? The most obvious approach would be to use spherical coordinates.
See problem 16 for that. I’ll use cylindrical coordinates instead. The element of volume isdV =r drdθdz, and
the integrals can be done a couple of ways.

∫

dV =

∫R

0

r dr

∫ 2 π

0

dθ

∫+√R (^2) −r 2
−

√

R^2 −r^2

dz=

∫+R

−R

dz

∫ 2 π

0

dθ

∫√R (^2) −z 2
0
r dr (15)
You can finish these now, see problem 17.
A Surface Charge Density
An example that appears in electrostatics: The surface charge density, dq/dA, on a sphere of radius R is
σ(θ,φ) =σ 0 sin^2 θcos^2 φ. What is the total charge on the sphere?