Calculus: Analytic Geometry and Calculus, with Vectors

694 Iterated and multiple integrals

turns out to be useful because the double integral can be compared with other
double integrals that are easily evaluated by use of polar coordinates. Let
D(h) be the quadrant of the circular disk consisting of points having polar coordi-
nates (p,¢) for which 0 5 p -< h and 0 S 0 < it/2 and let

(5) G(h) = fre

D(h) (=2+v2) dx dy.

Then, because the integrands are everywhere positive,

(6) G(h) 5 [F(h)]2 G(h \).

Writing (5) in terms of polar coordinates and evaluating the result by use of an

iterated integral gives

(7) G(h)= 1 JD(h) e-'*' p do dp f o, d( 0 e P'p dp

=fo/2 d4 [--fe °'];a0=

4

[1 - e"'].

Since F(h) > 0, this and (6) give

(8)^2 -v/!-- e " :_5

f0h

' dx 2 1 - e W.

Taking the limit as h --> oo gives the first formula in (1), and the second formula

in (1) follows from the first. The first formula in (2) is obtained from the second

formula in (1) by a change of variable; the trick is to set x = t/V2_ or and then

replace t by x in the new integral. The second formula in (2) is obtained by

integrating by parts and using the first formula in (2). Remark: The formulas

obtained by replacing x by x - M in (2) are important. The function c defined

by

_(Z-M)2

(9) I(x) _ e 2c

is the Gauss probability density function having mean (or average) M and standard

deviation or. In appropriate circumstances, the number

(10) faa 41(x) dx

is, when a < b, used for the probability that a number x (which could be the num-

ber of red corpuscles per cubic centimeter in your blood) lies between a and b.

The formula

M+.b M+ha - (z-M)' a

(11) f I(t) dt = f e 2`' dx =_ f ef't2 dt,

a M v`1r 0

which is proved by use of the substitution (x - M)/o- = t, facilitates calculations

of probabilities because the last member is tabulated as a function of X. Many

students of anthropology, medicine, education, agronomy, and other branches