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

Section 5.1 Areas and Distances 371
2 Definition The area A of the region S that lies under the graph of the continuous
function f is the limit of the sum of the areas of approximating rectangles:
A − lim
n l `
R n − lim
n l `
f f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dxg
It can be proved that the limit in Definition 2 always exists, since we are assuming that
f is continuous. It can also be shown that we get the same value if we use left endpoints:
3
A − lim
n l `
L n − lim
n l `
f f sx 0 d Dx 1 f sx 1 d Dx 1 ∙ ∙ ∙ 1 f sx n21 d Dxg
In fact, instead of using left endpoints or right endpoints, we could take the height of
the ith rectangle to be the value of f at any number x i * in the ith subinterval fx i21 , x i g.
We call the numbers x 1
*, x 2
*, . . . , x n * the sample points. Figure 13 shows approximating
rectangles when the sample points are not chosen to be endpoints. So a more general
expression for the area of S is
4
A − lim
n l `
f f sx 1
*d Dx 1 f sx 2
*d Dx 1 ∙ ∙ ∙ 1 f sx n * d Dxg
y
Îx
f(x i *)
0
a
⁄ x 2 ‹ x i-1 x i x n-1
b
x
FIGURE 13
x¡* x* x£* x i
*
x n
*
Note It can be shown that an equivalent definition of area is the following: A is the
unique number that is smaller than all the upper sums and bigger than all the lower sums.
We saw in Examples 1 and 2, for instance, that the area sA − 1 3d is trapped between
all the left approximating sums L n and all the right approximating sums R n . The function
in those examples, f sxd − x 2 , happens to be increasing on f0, 1g and so the lower sums
arise from left endpoints and the upper sums from right endpoints. (See Figures 8 and 9.)
In gen eral, we form lower (and upper) sums by choosing the sample points x i * so that
f sx i *d is the minimum (and maximum) value of f on the ith subinterval. (See Figure 14
and Exercises 7–8.)
y
FIGURE 14
Lower sums (short rectangles) and
upper sums (tall rectangles)
0
a
b
x
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