Calculus: Analytic Geometry and Calculus, with Vectors

680 Iterated and multiple integrals

can put these equations in the form

(13.461) x ffs S(x,y) dx dy =Ifs xS(x,y) dx dy,

y ffs S(x,y) dx dy=ffs yS(x,y)dx dy

or

IIs

xS(x,y) dx dy f f ys(x,y) dx dys

(13.462) x =

ffsS(x,y) dx dy A 3(x,y) dx dy

where the denominators are equal to the mass M of the lamina. It is

sometimes helpful to know that if, as in Figure 13.463, the line x = xo

is a line of symmetry of a homogeneous lamina, then lfI`2 = 0 and

hence x = xo In order to find the first moment MX('-', o of the lamina

about the y axis, it suffices to calculate the mass M of the lamina and

y

Figure 13.463

y

Figure 13.464

use the formula Mx = M.,('',. If, as in Figure 13.464, the lines x = xo

and y = yo are both lines of symmetry of a homogeneous lamina, then

x = xo and 5 = yo, so the centroid of the lamina is the point (xo,yo).

In case p = 2, the number M(p)g becomes

(13.47) M`2' =ffs(x - xo)25(x,y) dx dy,

the second moment or moment of inertia of the lamina about the line x = X.

When these things are being calculated and used in mechanics and else-

where, information concerning moments of inertia about parallel lines

(or axes) is very helpful. To obtain information of this nature, we let

x be the x coordinate of the centroid and use the simple identity

(x-xo)2 =[(x-x)+(x-x0)2

_ (x - g) l -F (x - xo) 2 -}- 2(9 - xo) (x - x)

to put (13.47) in the form

(13.471) M`2Z_ZQ=ffs (x- x)2S(x,y) dx dy + (x - xo)2f f, S(x,y) dx dy

• 2(9 - xo) ffs (x - z)s(x,y) dx dy.

The first term in the right member is M`2', the moment of inertiaof the