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

discovery project Area Functions 391
67. Which of the integrals y 2 arctan x dx, 1 y2 arctan sx 1
dx, and
y 2 1
arctanssin xd dx has the largest value? Why?
68. Which of the integrals y 0.5
0 cossx 2 d dx, y 0.5
0 cos sx dx is larger?
Why?
69. Prove Property 3 of integrals.
70. (a) If f is continuous on fa, bg, show that
u y b
f sxd dx
a u < y b
| f sxd | dx
a
[Hint: 2| f sxd | < f sxd < | f sxd | .]
(b) Use the result of part (a) to show that
u y 2
f sxd sin 2x dx
0
u < y 2
| f sxd | dx
0
71. Let f sxd − 0 if x is any rational number and f sxd − 1 if x is
any irrational number. Show that f is not integrable on f0, 1g.
72. Let f s0d − 0 and f sxd − 1yx if 0 , x < 1. Show that f
is not integrable on f0, 1g. [Hint: Show that the first term
in the Riemann sum, f sx 1
*d Dx, can be made arbitrarily
large.]
73–74 Express the limit as a definite integral.
73. lim
n l ` o n
i−1
1
74. lim
n l ` n on
i−1
i 4
n 5 [Hint: Consider f sxd − x 4 .]
1
1 1 siynd 2
75. Find y 2 1 x 22 dx. Hint: Choose x i * to be the geometric mean
of x i21 and x i (that is, x i * − sx i21x i ) and use the identity
1
msm 1 1d − 1 m 2 1
m 1 1
discovery Project
area functions
1. (a) Draw the line y − 2t 1 1 and use geometry to find the area under this line, above the
t-axis, and between the vertical lines t − 1 and t − 3.
(b) If x . 1, let Asxd be the area of the region that lies under the line y − 2t 1 1
between t − 1 and t − x. Sketch this region and use geometry to find an expression
for Asxd.
(c) Differentiate the area function Asxd. What do you notice?
2. (a) If x > 21, let
Asxd − y x s1 1 t 2 d dt
21
Asxd represents the area of a region. Sketch that region.
(b) Use the result of Exercise 5.2.28 to find an expression for Asxd.
(c) Find A9sxd. What do you notice?
(d) If x > 21 and h is a small positive number, then Asx 1 hd 2 Asxd represents the
area of a region. Describe and sketch the region.
(e) Draw a rectangle that approximates the region in part (d). By comparing the areas of
these two regions, show that
Asx 1 hd 2 Asxd
< 1 1 x 2
h
(f) Use part (e) to give an intuitive explanation for the result of part (c).
;
3. (a) Draw the graph of the function f sxd − cossx 2 d in the viewing rectangle f0, 2g
by f21.25, 1.25g.
(b) If we define a new function t by
tsxd − y x
cosst 2 d dt
0
then tsxd is the area under the graph of f from 0 to x [until f sxd becomes negative, at
which point tsxd becomes a difference of areas]. Use part (a) to determine the value
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