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

370 Chapter 5 Integrals
The width of the interval fa, bg is b 2 a, so the width of each of the n strips is
Dx − b 2 a
n
These strips divide the interval fa, bg into n subintervals
fx 0 , x 1 g, fx 1 , x 2 g, fx 2 , x 3 g, . . . , fx n21 , x n g
where x 0 − a and x n − b. The right endpoints of the subintervals are
x 1 − a 1 Dx,
x 2 − a 1 2 Dx,
x 3 − a 1 3 Dx,
∙
∙
∙
Let’s approximate the ith strip S i by a rectangle with width Dx and height f sx i d, which
is the value of f at the right endpoint (see Figure 11). Then the area of the ith rectangle
is f sx i d Dx. What we think of intuitively as the area of S is approximated by the sum of
the areas of these rectangles, which is
R n − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx
y
Îx
f(x i )
FIGURE 11
0
a
⁄ x 2 ‹
x i-1
x i
b
x
Figure 12 shows this approximation for n − 2, 4, 8, and 12. Notice that this approximation
appears to become better and better as the number of strips increases, that is, as
n l `. Therefore we define the area A of the region S in the following way.
y
y
y
y
0 a ⁄
(a) n=2
FIGURE 12
b x 0 a ⁄ x 2 ‹ b x 0 a
b x 0 a
b
(b) n=4
(c) n=8
(d) n=12
x
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