Calculus: Analytic Geometry and Calculus, with Vectors

4.8 Moments and centroids in Ez and E, 271

val (or the sum of the lengths of the intervals) in which the line having the

equation y = y* intersects the lamina. In case p = 0, the pth moment

is the mass of thelamina. In mechanics and some other places, thesec-

ond moment is called the moment of inertia.

While the facts can be established only by considering the different

rectangular coordinate systems in the plane of the lamina, the lamina

itself determines a point in the plane of the lamina that is called the

centroid of the lamina. With reference to the particular coordinate

system which we have chosen, the x coordinate of this centroid is the

number x for which Ma_t = 0 when E = x. Thus

(4.84)

and it follows that

S

fab

(x - 9)f(x) dx = 0

S bf b x

(4.841) Mx = S

fb

xf(x) dx, x = a

f(x) dx

f6

af(x) dx

where M, the denominator in the second formula, is the mass of the

lamina. Similar formulas suffice to determine the y coordinate 9 of the

centroid. For example,

(4.842) M9 = S

fcd

yg(y) dy

The centroid of a lamina has an important physical property. If the
lamina is in a plane perpendicular to the direction of the forces in a
uniform parallel force field, then the lamina will balance upon each line
(or knife-edge) which passes through the centroid and will balance upon a
pin placed at the centroid. It follows that if L is a line of symmetry of a
lamina, then the centroid lies on L. Moreover, if a point P is a center of
symmetry of a lamina, then the centroid is P.
We now turn our attention to the three-dimensional world which con-
tains, in addition to cubes and spherical balls, so many distractions that
relatively few of its inhabitants assimilate substantial information about
nonmeasurable sets in E3. To keep these complicated and paradoxical
sets out of our gardens, we shouldt (and therefore do) start with a set
S in E3 which is assumed to possess positive volume V. In order to be
able to use Riemann integrals, we assume that wherever we introduce an
x, y, z coordinate system in E3, there will be numbers a and b for which our
set S lies between the planes having the equations x = a and x = b.
We assume that, for each t for which a < t S b, the plane having the
equation x = t intersects S in a section having area which we denote by
il(t). In many applications this area function is continuous. To be
t This is another situation in which we can be kept on the path of rectitude by knowledge
of the contents of Appendix 2 at the end of this book.