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

Section 5.1 Areas and Distances 369
From Figures 8 and 9 it appears that, as n increases, both L n and R n become better and
bet ter approximations to the area of S. Therefore we define the area A to be the limit of
the sums of the areas of the approximating rectangles, that is,
TEC In Visual 5.1 you can create
pictures like those in Figures 8 and 9
for other values of n.
A − lim
n l `
R n − lim
n l `
L n − 1 3
y
y
y
n=10 R¡¸=0.385
n=30 R£¸Å0.3502
n=50 R∞¸=0.3434
0 1 x 0
1 x 0
1 x
FIGURE 8 Right endpoints produce upper sums because f sxd − x 2 is increasing.
y
y
y
n=10 L¡¸=0.285
n=30 L£¸Å0.3169
n=50 L∞¸=0.3234
0
1
x
0
1
x
0 1 x
FIGURE 9 Left endpoints produce lower sums because f sxd − x 2 is increasing.
Let’s apply the idea of Examples 1 and 2 to the more general region S of Figure 1.
We start by subdividing S into n strips S 1 , S 2 , . . . , S n of equal width as in Figure 10.
y
y=ƒ
S¡ S S£ S i S n
FIGURE 10
0
a
⁄ x 2 ‹
. . . x i-1 . . . x n-1
x i
b
x
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