Calculus: Analytic Geometry and Calculus, with Vectors

13.4 Applications of double and iterated integrals 681

lamina about the line through the centroid parallel to the linex = X.
The second term is (x - xo)2M, where M is the mass of the lamina.
The third term is 0 because the integral is the first moment of a lamina
about a line through its centroid. Thus (13.471) reduces to the impor-
tant formula

(13.472) Ms? = Mx'= -F (z- xo)2M.

This gives the following parallel axis theorem.

Theorem 13.48 The moment of inertia ofa lamina about a line is equal

to the sum of two terms, one being the moment of inertia of the lamina about

the parallel line through the centroid and the other being the product of the

mass M of the lamina and the square of the distance between the two lines

Up to the present time, we have considered

only moments of plane laminas about lines in y

the planes of the laminas. The second moment

or moment of inertia of a lamina about a line

L perpendicular to the plane of the lamina is

!

called the polar moment of inertia of the lamina

(xo.yo)

s

about the line L. As before, let the lamina

cover a set S in the xy plane and let L be the

Figure 13.481

line in E3 having the equations x = xo, y = yo. Letting AS or Ax Ay be

the area of a part of the set S which contains the point (x,y) and letting

S(x,y) denote the density of the lamina at the point (x,y), we use the number

(13.482) [(x - xo)2 + (Y - Yo)293(x,Y) Ax AY

as an approximation to the polar moment of inertia about L of the part

of the lamina. The polar moment of inertia about L of the whole lamina

may be denoted by the symbol MXO It is defined by the formula

(13.483) M=2)xo.vmva = Jim J[(x - xo)2 + (y - yo)2)3(x,Y) Ax Ay

or

//'((

(13.484) M`2 fs[(x - xo)2 + (y - Yo)2ls(x,Y) dx dy.

Comparing this with the formula (13.47) for MIA and the corresponding

formula for My_ogives the formula

(13.485) Mx2='xo'Y-YG =M.(21o +Mv2>vo'

which says that the polar moment of inertia of the lamina about the

line x = xo, y = yo perpendicular to the lamina is equal to the sum ofthe

moments of inertia of the lamina about the two lines x = xoand y = yo

in the plane of the lamina. Finally, we note that the parallel axis

theorem, Theorem 13.48, holds for polar moments of inertia aswell as

for moments of inertia about lines in the plane of a lamina.