270 Integrals
the set of points (x,y) for which a < x S b and fl(x) 5 y < f2(x), then
everything is quite simple and f(x) = f2(x) - fi(x). Our next step isto
suppose that the points (x,y) of R are points of a material body (much like
a sheet of paper or a sheet of copper) that has been compressed into a
plane in such a way that, for some positive constant S, a part of the com-
pressed material body has mass S AR if that part occupies a part of the
region R having area AR. The compressed body is called a lamina, and
it is a homogeneous lamina because the areal density (mass per unit area)
has the same constant value S at all places in the lamina.
Letting p be an integer which is either 0 or positive, we proceed to define
the number Mz'-')e, the pth moment of the lamina about the line having
the equation x = , by a formula from which it can be calculated. Fol-
lowing the method of Section 4.7, we make a partition P of the interval
a 5 x < b into subintervals. For each k, the lines having the equations
x = xk_1 and x = Xk have between them a part of the lamina that can be
called a strip parallel to the line having the equation x Supposing
as usual that xk-1 < xk* < xk, we use the number
(4.82) f (xk) LIxkas an approximation to the area of the strip and accordingly use the
number(4.821) Sf (xk) Lxk
as an approximation to the mass of the strip. When the norm of P is
small, all points of the strip lie at about the same distance Ixl* - El from
the line having the equation x = t, and multiplying the above mass by
(xx - )P should therefore give a good approximation to the pth moment
of the strip about the line having the equation x =. The Riemann sum(4.822) SZ(xk - )pf(xk) .Xk
should then be a good approximation to the moment of the whole lamina.
Since the Riemann sums have a limit which is the Riemann integral in
the right member of the formula(4.83) Mgt = S f ab (x - E)Pf(x) dx,our work motivates the definition by which the required moment is
defined by this formula.
The number the pth moment of the lamina about the line
having the equation y = rl, is defined by the analogous formula(4.831) M = 8Jd (y - 0)Pg(y) dy,
where c and d are numbers such that the lamina lies between the lines
having the equation y = c and y = d and g(y*) is the length of the inter-