Calculus: Analytic Geometry and Calculus, with Vectors

13.6 Triple integrals; rectangular coordinates 699

Thus (13.653) and

(13.654) Ox

a:(x> dy

9,(x) )PS x z A

1,(.,v) x - xo (,y, )

are approximations to the required moment of one slab. Finally,

(13.655) r oz(x)dy

f,(x,v)

OxJa),(x ff,(x,y) (x - xo)DS(x,y,z) A

is an approximation to the required moment of the whole solid and
replacing this by its limit gives the formula

(13.656) M(P) = °2dx f,(°,(x)1x)dy J_(xn)

a, g fi(x.',) (x - xo)p3(x,y,z) A

for the required pth moment of the solid about the plane x = xo.
Remarks very similar to those following (13.453) can now be made.
Formulas for the pth moments of the solid about the planes y= yo
and z = zo are obtained by replacing the factor (x- xo)P by the factors
(y - yo)P and (z - zo)P in the above derivations. In case p = 0, the

factors (x - xo)P, (y - yo)P, (z - zo)" are all 1 and the numbers M=JX
M(P'EO are all equal to the mass M of the solid. Thus

(13.66) M = fffs 6(x,y,z) dx dy dz,

and we can replace this triple integral by iterated integrals. Formulas
very similar to (13.46) show that the formulas M`(" = 0, 0,
M=i>r = 0 which determine the centroid (z,y,z) of S can be put in the

form

(13.67) x =

f f fs xS(x,y,z) dx dy A f ffsya(x,y,z) dx dy dz

S(x,y,z) dx dy A

y

fffs S(x,y,z) dx dy A

NS

fffszS(x,y,z) dx dy dz

fffs S(x,y,z) dx dy dz

where the denominators are equal to the mass of the solid.
For some purposes, the polar moment of inertia of a solid about a line
L is of importance. When the line L is the line having the equations
x = xo, y = yo, we may denote the polar moment of inertia about L
by the symbol or Ix-x,,,,J and work out the formula

(13.68) MX2)xo,7/yo =Ix=za,Y YD

= fffs[(x-xo)2 + (y - yo)2JS(x,v,z) dx dy dz

which is analogous to (13.484). The formula

(13.681) M`x2'x,.,,_v0 = Mz2_'xO +Mv='o