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(b) If f is an odd function (Section 7.3), show that, for any a > 0,
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If u = x, then du = dx. Hence, for any integrable function f (x),
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f ( u) du =
f ( u) du
CHAP. 31]
THE FUNDAMENTAL THEOREM OF CALCULUS
notation
Renaming the variable in a de nite integral does not affect the value of the integral:
g(x) dx =
g(t) dt =
g( ) d =
Thus, changing u to x,
0 a a
f (x) dx =
f ( x) dx
and so
f (x) dx = = =
0 a a 0 a 0
f (x) dx + f ( x) dx +
f (x) dx
[by Theorem 30.4] [by (1)] [by Theorem 30.3]
f (x) dx
(f (x) + f ( x)) dx
(a) For an even function, f (x) + f ( x) = 2f (x), whence,
a a
f (x) dx =
2f (x) dx = 2
f (x) dx
(b) For an odd function, f (x) + f ( x) = 0, whence,
a a
f (x) dx =
0 dx = 0
dx = 0
notation
One usually writes
instead of
1 dx
31.9 (a) Let f (x) 0 on [a, b], and let [a, b] be divided into n equal parts, of length points x1 , x2 , . . . , xn 1 [see Fig. 31-2(a)]. Show that
x = (b a)/n, by means of
f (x) dx
f (a) + 2
f (xi ) + f (b)
trapezoidal rule
(b) Use the trapezoidal rule, with n = 10, to approximate
x 2 dx
(= 0.333 . . .)
(a) The area in the strip over the interval [xi 1 , xi ] is approximately the area of trapezoid ABCD in Fig. 31-2(b), which is x [ f (xi 1 ) + f (xi )] 2 geometry The area of a trapezoid of height h and bases b1 and b2 is 1 h(b1 + b2 ) 2
THE FUNDAMENTAL THEOREM OF CALCULUS
[CHAP. 31
where we understand x0 = a, xn = b. The area under the curve is then approximated by the sum of the trapezoidal areas,
f (x) dx
x {( f (x0 ) + f (x1 )) + ( f (x1 ) + f (x2 )) + + ( f (xn 1 ) + f (xn ))} 2 n 1 x f (xi ) + f (b) = f (a) + 2 2
Fig. 31-2
(b) By the trapezoidal rule, with n = 10, a = 0, b = 1, x = 1/10, xi = i/10, 9 9 1 1 2 1 2 i2 0 +2 + 12 = x 2 dx i2 + 1 20 100 20 100 0
i=1 i=1
1 2 (285) + 1 [by arithmetic or Problem 30.12(a, ii)] = 20 100 1 285 = + = 0.285 + 0.050 = 0.335 1000 20 whereas the exact value is 0.333 . . . 1
Supplementary Problems
31.10 Use the fundamental theorem to compute the following de nite integrals:
(a) (d)
1 16 1
(3x 2 2x + 1) dx x 3/2 dx
cos x dx 2 x dx x
sec2 x dx x 2 6x + 9 dx
(f )
31.11 Calculate the areas under the graphs of the following functions, above the x-axis and between the two indicated values a and b of x. [In part (g), the area below the x-axis is counted negative.] (a) f (x) = sin x a = , b = 6 3
1 (c) f (x) = (a = 1, b = 8) 3 x
(b) f (x) = x 2 + 4x (a = 0, b = 3) (d) f (x) = 4x + 1 (a = 0, b = 2)
f has a continuous second derivative, it can be shown that the error in approximating a f (x) dx by the trapezoidal rule is at most ((b a)/12n2 )M, where M is the maximum of f (x) on [a, b] and n is the number of subintervals.
1 When
CHAP. 31]
THE FUNDAMENTAL THEOREM OF CALCULUS
(e) f (x) = x 2 3x (a = 3, b = 5) (g) f (x) = x 2 (x 3 2) (a = 1, b = 2) 31.12 Compute the following de nite integrals:
(f ) f (x) = sin2 x cos x a = 0, b = 2 (h) f (x) = 4x x 2 (a = 0, b = 3)
cos x sin x dx
/2
(b) (e) (h) (k)
1 1
tan x sec2 x dx x + 2 x 2 dx x dx dx
(c) (f )
(d) (g)
0 15 3 2
sin x + 1 cos x dx
0 2
1 5 2 8
3x 2 2x + 3 (3x 1) dx x 3 4 x 5 dx
x 2 9 x 3 dx
(j) (m)
1 /8 0
| x 1| dx sec2 2x tan3 2x dx
0 (2x 2 + 1)3 2 x 7 2x + 1 1
(i) (l)
4x 3
x dx 0 (x + 1)3/2 0 (x + 2) x + 3 dx
[Hint: Apply Theorem 30.4 to part (j).] 31.13 Compute the average value of each of the following functions on the given interval: (a) f (x) = 3 x on [0, 1] (b) f (x) = sec2 x on 0, 4 (c) f (x) = x 2 2x 1 on [ 2, 3] (d) f (x) = sin x + cos x on [0, ]
31.14 Verify the mean-value theorem for integrals in the following cases: (a) f (x) = x + 2 on [1, 2] (b) f (x) = x 3 on [0, 1] (c) f (x) = x 2 + 5 on [0, 3]
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