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

372 Chapter 5 Integrals
This tells us to
end with i=n.
This tells us
to add.
This tells us to
start with i=m.
n
µ f(x i ) Îx
i=m
We often use sigma notation to write sums with many terms more compactly. For
instance,
o n
f sx i d Dx − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx
i−1
So the expressions for area in Equations 2, 3, and 4 can be written as follows:
If you need practice with sigma notation,
look at the examples and try some of
the exercises in Appendix E.
A − lim
n l ` o n
f sx i d Dx
i−1
A − lim
n l ` o n
f sx i21 d Dx
i−1
A − lim
n l ` o n
f sx i
*d Dx
i−1
We can also rewrite Formula 1 in the following way:
o n nsn 1 1ds2n 1 1d
i 2 −
i−1
6
Example 3 Let A be the area of the region that lies under the graph of f sxd − e 2x
between x − 0 and x − 2.
(a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit.
(b) Estimate the area by taking the sample points to be midpoints and using four sub -
intervals and then ten subintervals.
SOLUTION
(a) Since a − 0 and b − 2, the width of a subinterval is
Dx − 2 2 0
n
− 2 n
So x 1 − 2yn, x 2 − 4yn, x 3 − 6yn, x i − 2iyn, and x n − 2nyn. The sum of the areas of
the approximating rectangles is
According to Definition 2, the area is
R n − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx
− e 2x 1 Dx 1 e2x 2 Dx 1 ∙ ∙ ∙ 1 e2xn Dx
− e 22ynS 2 nD 1 e 24ynS 2 nD 1 ∙ ∙ ∙ 1 e 22nynS 2 nD
2
A − lim R n − lim
n l ` n l ` n se22yn 1 e 24yn 1 e 26yn 1 ∙ ∙ ∙ 1 e 22nyn d
Using sigma notation we could write
2
A − lim
n l ` n on e 22iyn
i−1
It is difficult to evaluate this limit directly by hand, but with the aid of a computer algebra
system it isn’t hard (see Exercise 30). In Section 5.3 we will be able to find A more
easily using a different method.
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