Page 113 - Schaum's Outline of Theory and Problems of Advanced Calculus

104 INTEGRALS [CHAP. 5

MEAN VALUE THEOREMS FOR INTEGRALS
5.9. Given the right triangle pictured in Fig. 5-6: (a) Find the
average value of h.(b)At what point does this average value
occur? (c) Determine the average value of
1
f ðxÞ¼ sin 1 x; 0 @ x @ . (Use integration by parts.)
2
2
(d) Determine the average value of f ðxÞ¼ cos x; 0 @ x @ .
2
H
x.According to the mean value theorem for integrals,
B Fig. 5-6
(a) hðxÞ¼
the average value of the function h on the interval ½0; BŠ is
1 ð B H H
A ¼ xdx ¼
B 0 B 2
(b) The point, ,at which the average value of h occurs may be obtained by equating f ð Þ with that average
H H B
value, i.e., . Thus, ¼ .
B ¼ 2 2
FUNDAMENTAL THEOREM OF THE CALCULUS
ð x
f ðtÞ dt where f ðxÞ is continuous in ½a; bŠ, prove that F ðxÞ¼ f ðxÞ.
0
5.10. If FðxÞ¼
a
1 ð xþh ð x 1 ð xþh
Fðx þ hÞ FðxÞ
f ðtÞ dt
h ¼ h a f ðtÞ dt a f ðtÞ dt ¼ h x
between x and x þ h
¼ f ð Þ
by the first mean value theorem for integrals (Page 93).
Then if x is any point interior to ½a; bŠ,
F ðxÞ¼ lim Fðx þ hÞ FðxÞ ¼ lim f ð Þ¼ f ðxÞ
0
h!0 h h!0
since f is continuous.
If x ¼ a or x ¼ b,weuse right- or left-hand limits, respectively, and the result holds in these cases as
well.

5.11. Prove the fundamental theorem of the calculus, Part 2 (Pages 94 and 95).
By Problem 5.10, if FðxÞ is any function whose derivative is f ðxÞ,wecan write
ð x
f ðtÞ dt þ c
FðxÞ¼
a
where c is any constant (see last line of Problem 22, Chapter 4).
b b
ð ð
f ðtÞ dt þ FðaÞ or f ðtÞ dt ¼ FðbÞ FðaÞ.
Since FðaÞ¼ c,it follows that FðbÞ¼
a a
x
ð
f ðtÞ dt is continuous in ½a; bŠ.
5.12. If f ðxÞ is continuous in ½a; bŠ,prove that FðxÞ¼
a
If x is any point interior to ½a; bŠ,thenasinProblem 5.10,
lim Fðx þ hÞ FðxÞ¼ lim hf ð Þ¼ 0
h!0 h!0
and FðxÞ is continuous.
If x ¼ a and x ¼ b,weuse right- and left-hand limits, respectively, to show that FðxÞ is continuous at
x ¼ a and x ¼ b.