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f (x) dx 0 [by Theorem 30.3] g(x) dx
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[by (b)] [by Theorem 30.2] [by Problem 30.1]
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f (x) dx M
1 dx
m(b a)
f (x) dx M(b a)
Supplementary Problems
30.4 Evaluate:
8 dx
5x 2 dx
(x 2 + 4) dx
[Hint: Use Problems 30.1 and 30.2.]
CHAP. 30]
THE DEFINITE INTEGRAL
30.5 For the function f graphed in Fig. 30-9, express
f (x) dx in terms of the areas A1 , A2 , A3 .
Fig. 30-9
n(n + 1) b2 . You may assume the formula 1 + 2 + + n = proved in Problem 30.12(a). 2 2 0 Check your result by using the standard formula for the area of a triangle. [Hint: Divide the interval [0, b] into n equal subintervals, and choose xi = ib/n, the right endpoint of the ith subinterval.]
30.6 (a) Show that
x dx =
(b) Show that
x dx =
b2 a 2 . [Hint: Use (a) and Theorem 30.4.] 2
(c) Evaluate
5x dx. [Hint: Use Theorem 30.2 and (b).]
30.7 Show that the equation of Theorem 30.4,
f (x) dx +
f (x) dx =
f (x) dx
holds for any numbers a, b, c, such that the two de nite integrals on the left can be de ned in the extended sense. [Hint: Consider all six arrangements of distinct a, b, c: a < b < c, a < c < b, b < a < c, b < c < a, c < a < b, c < b < a. Also consider the cases where two of the numbers are equal or all three are equal.] 30.8 Show that 1
2 2 1
x 3 dx 8. [Hint: Use Problem 30.3(c).]
30.9 (a) Find
4 x 2 dx by using a formula of geometry. [Hint: What curve is the graph of y =
4 x 2 ]
(b) From part (a) infer that 0 4. (Much closer estimates of are obtainable this way.) 30.10 Evaluate:
i=1 3
(3i 1) sin j 6
k=0 5
(3k 2 + 4) f 1 n if f (x) = 1 x
30.11 If f is continuous on [a, b], f (x) 0 on [a, b], and f (x) > 0 for some x in [a, b], show that
f (x) dx > 0
[Hint: By continuity, f (x) > K > 0 on some closed interval inside [a, b]. Use Theorem 30.4 and Problem 30.3(c).]
THE DEFINITE INTEGRAL
[CHAP. 30
30.12 (a) Use mathematical induction (see Problem 12.2) to prove: n(n + 1)(2n + 1) n(n + 1) (ii) 12 + 22 + + n2 = (i) 1 + 2 + + n = 2 6 (b) By looking at the cases when n = 1, 2, 3, 4, 5, guess a formula for 13 + 23 + + n3 and then prove it by mathematical induction. [Hint: Compare the values of formula (i) in part (a) for n = 1, 2, 3, 4, 5.] 30.13 If the graph of f between x = 1 and x = 5 is as shown in Fig. 30-10, evaluate
f (x) dx.
Fig. 30-10
30.14 Let f (x) = 3x + 1 for 0 x 1. If the interval [0, 1] is divided into ve subintervals of equal length, what is the smallest corresponding Riemann sum (30.1)
The Fundamental Theorem of Calculus
31.1 CALCULATION OF THE DEFINITE INTEGRAL We shall develop a simple method for calculating
f (x) dx
a method based on a profound and surprising connection between differentiation and integration. This connection, discovered by Isaac Newton and Gottfried von Leibniz, the co-inventors of calculus, is expressed in the following: Theorem 31.1: Let f be continuous on [a, b]. Then, for x in [a, b],
f (t) dt
is a function of x such that
f (t) dt
= f (x)
A proof may be found in Problem 31.5. Now for the computation of the de nite integral, let F(x) = f (x) dx denote some known anti-derivative of x f (x) (for x in [a, b]). According to Theorem 31.1, the function a f (t)dt is also an anti-derivative of f (x). Hence, by Corollary 29.2,
f (t) dt = F(x) + C
for some constant C. When x = a, 0=
f (t)dt = F(a) + C
C = F(a)
Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
THE FUNDAMENTAL THEOREM OF CALCULUS
[CHAP. 31
Thus, when x = b,
f (t) dt = F(b) F(a)
and we have proved: Theorem 31.2 (Fundamental Theorem of Calculus): Let f be continuous on [a, b], and let F(x) =
f (x) dx. Then,
f (x) dx = F(b) = F(a)
notation
The difference F(b) F(a) will often be denoted by F(x)]b , and the fundamental theorem notated as a
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