James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

384 Chapter 5 Integrals
y
1
0
FIGURE 9
y
1
y=x-1
y= œ„„„„„ 1-≈
or
≈+¥=1
A¡
(3, 2)
x
Example 4 Evaluate the following integrals by interpreting each in terms of areas.
(a) y 1
s1 2 x 2 dx (b) y 3
sx 2 1d dx
0 0
SOLUTION
(a) Since f sxd − s1 2 x 2 > 0, we can interpret this integral as the area under the
curve y − s1 2 x 2 from 0 to 1. But, since y 2 − 1 2 x 2 , we get x 2 1 y 2 − 1, which
shows that the graph of f is the quarter-circle with radius 1 in Figure 9. Therefore
y 1
0 s1 2 x 2 dx − 1 4 s1d2 − 4
(In Section 7.3 we will be able to prove that the area of a circle of radius r is r 2 .)
(b) The graph of y − x 2 1 is the line with slope 1 shown in Figure 10. We compute
the integral as the difference of the areas of the two triangles:
y 3
sx 2 1d dx − A 1 2 A 2 − 1 2 s2 ∙ 2d 2 1 2 s1 ∙ 1d − 1.5
n
0
0
_1
A 1
FIGURE 10
3
x
The Midpoint Rule
We often choose the sample point x i * to be the right endpoint of the ith subinterval
because it is convenient for computing the limit. But if the purpose is to find an approximation
to an integral, it is usually better to choose x i * to be the midpoint of the interval,
which we denote by x i . Any Riemann sum is an approximation to an integral, but if we
use midpoints we get the following approximation.
TEC Module 5.2 / 7.7 shows how the
Midpoint Rule estimates improve as n
increases.
Midpoint Rule
where
and
y b
f sxd dx < o n
f sx i d Dx − Dx f f sx 1 d 1 ∙ ∙ ∙ 1 f sx n dg
a
i−1
Dx − b 2 a
n
x i − 1 2 sx i21 1 x i d − midpoint of fx i21 , x i g
Example 5 Use the Midpoint Rule with n − 5 to approximate y 2
1
1
x dx.
SOLUTION The endpoints of the five subintervals are 1, 1.2, 1.4, 1.6, 1.8, and 2.0,
so the midpoints are 1.1, 1.3, 1.5, 1.7, and 1.9. The width of the subintervals is
Dx − s2 2 1dy5 − 1 5 , so the Midpoint Rule gives
y 2
1
1
dx < Dx f f s1.1d 1 f s1.3d 1 f s1.5d 1 f s1.7d 1 f s1.9dg
x
− S 1 1
5 1.1 1 1
1.3 1 1
1.5 1 1
1.7 1.9D 1 1
< 0.691908
Since f sxd − 1yx . 0 for 1 < x < 2, the integral represents an area, and the approxi -
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