Page 101 - Schaum's Outline of Theory and Problems of Advanced Calculus

92 INTEGRALS [CHAP. 5

ð b ð c ð b
3. f ðxÞ dx ¼ f ðxÞ dx þ f ðxÞ dx provided f ðxÞ is integrable in ½a; cŠ and ½c; bŠ.
a a c

ð b ð a
4. f ðxÞ dx ¼ f ðxÞ dx
a b
a
ð
5. f ðxÞ dx ¼ 0
a
6. If in a @ x @ b, m @ f ðxÞ @ M where m and M are constants, then

ð b
mðb aÞ @ f ðxÞ dx @ Mðb aÞ
a
7. If in a @ x @ b, f ðxÞ @ gðxÞ then
b b
ð ð
f ðxÞ dx @ gðxÞ dx
a a

ð b ð b

8. f ðxÞ dx @ j f ðxÞj dx if a < b

a a
MEAN VALUE THEOREMS FOR INTEGRALS
As in differential calculus the mean value theorems listed below are existence theorems. The first
one generalizes the idea of finding an arithmetic mean (i.e., an average value of a given set of values) to a
continuous function over an interval. The second mean value theorem is an extension of the first one
that defines a weighted average of a continuous function.
By analogy, consider determining the arithmetic mean (i.e., average value) of temperatures at noon
for a given week. This question is resolved by recording the 7 temperatures, adding them, and dividing
by 7. To generalize from the notion of arithmetic mean and ask for the average temperature for the
week is much more complicated because the spectrum of temperatures is now continuous. However, it
is reasonable to believe that there exists a time at which the average temperature takes place. The
manner in which the integral can be employed to resolve the question is suggested by the following
example.
Let f be continuous on the closed interval a @ x @ b. Assume the function is represented by the
correspondence y ¼ f ðxÞ,with f ðxÞ > 0. Insert points of equal subdivision, a ¼ x 0 ; x 1 ; ... ; x n ¼ b.
Then all x k ¼ x k x k 1 are equal and each can be designated by x. Observe that b a ¼ n x.
Let k be the midpoint of the interval x k and f ð k Þ the value of f there. Then the average of these
functional values is
n
½ f ð 1 Þþ þ f ð n ފ x 1 X
f ð 1 Þþ þ f ð n Þ
f ð Þ
n b a b a
¼
¼
k¼1
This sum specifies the average value of the n functions at the midpoints of the intervals. However,
we may abstract the last member of the string of equalities (dropping the special conditions) and define
n
1 X 1 ð b
lim f ðxÞ dx
n!1 b a f ð Þ ¼
k¼1 b a a
as the average value of f on ½a; bŠ.