CalculusVolume2-SASG-01-01
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 1
Integration
1.1. Approximating Areas
Section Exercises
1.
State whether the given sums are equal or unequal.
a.
and b.
and c.
and d.
and Answer: a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting
. d. They are equal; the first sum factors the terms of the second.
In the following exercises, use the rules for sums of powers of integers to compute the sums.
3.
Answer: Suppose that and . In the following exercises, compute the sums.
5.
Answer: 7.
Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
9.
Answer: 11.
Answer: Let denote the left-endpoint sum using n
subintervals and let denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.
13.
R
4
for on Answer: 15.
R
6
for on Answer: 17.
L
4
for on Answer: 19.
L
8
for on Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
21.
Compute the left and right Riemann sums—
L
6
and R
6
, respectively—for
on . Compute their average value and compare it with the area under the graph of f
.
Answer: . The graph of f
is a triangle with area 9.
23.
Compute the left and right Riemann sums—
L
6
and R
6
, respectively—for
on and compare their values.
Answer: . They are equal.
Express the following endpoint sums in sigma notation but do not evaluate them.
25.
L
10
for on Answer: 27.
R
100
for on Answer: In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve on the given interval better approximated by the left Riemann sum or right Riemann sum? If the two agree, say “neither.”
29.
[T]
L
100
and R
100
for on the interval Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
, . The plot shows that the left Riemann sum is an underestimate because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area
lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
31.
[T]
L
100
and R
100
for on the interval Answer: , . The left endpoint sum is an underestimate because the function is increasing. Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates.
33.
[T]
L
100
and R
100
for on the interval Answer: , . The plot shows that the left Riemann sum is an underestimate because the function is increasing. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.
35.
Let denote the total rainfall in Portland on the j
th day of the year in 2009. Interpret
.
Answer: The sum represents the cumulative rainfall in January 2009.
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
37.
To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds mi to his daily routine each week, what is the total mileage on Joe’s shoes after 25 weeks?
Answer: The total mileage is .
39.
The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to 2010.
Approximate 20-Year Sea Level Increases, 1870–1990
Starting Year
20-Year Change
1870
0.3
1890
1.5
1910
0.2
1930
2.8
1950
0.7
1970
1.1
1990
1.5
Answer: Add the numbers to get 8.1-in. net increase.
41.
The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010.
Annual Percentage Growth of U.S. Population, 2000–2009
Year
% Change/Year
2000
1.12
2001
0.99
2002
0.93
2003
0.86
2004
0.93
2005
0.93
2006
0.97
2007
0.96
2008
0.95
2009
0.88
(
Hint:
To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.)
Answer: 309,389,957
OpenStax Calculus Volume 2
Student Answer and Solution Guide
In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L
8
.
43.
Answer: 45.
Answer: 47.
[T]
Use a computer algebra system to compute the Riemann sum, L
N
, for for on .
Answer: , ,
OpenStax Calculus Volume 2
Student Answer and Solution Guide
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R
N
and L
N
for . How do these estimates compare with the exact answers, which you can find via geometry?
49.
[T]
on the interval Answer: , , , , , and . By symmetry of the
graph, the exact area is zero.
In the following exercises, use a calculator or a computer program to evaluate the endpoint sums R
N
and L
N
for .
51.
[T]
on the interval , which has an exact area of Answer: , , , , , 53.
Explain why, if and f
is increasing on , that the left endpoint estimate is a
lower bound for the area below the graph of f
on .
Answer: If is a subinterval of under one of the left-endpoint sum rectangles, then the area of the rectangle contributing to the left-endpoint estimate is . But,
for , so the area under the graph of f between c
and d
is plus the area below the graph of f
but above the horizontal line segment at height , which is
positive. As this is true for each left-endpoint sum interval, it follows that the left Riemann sum is less than or equal to the area below the graph of f
on .
55.
Show that, in general, .
Answer: and
. The left sum has a term corresponding to and the right sum has a term corresponding to . In , any term corresponding to occurs once with a plus sign and once with a minus sign, so each such term cancels and one is left with .
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
OpenStax Calculus Volume 2
Student Answer and Solution Guide
57.
For each of the three graphs:
a.
Obtain a lower bound for the area enclosed by the curve by adding the areas of the
squares enclosed completely by the curve.
b.
Obtain an upper bound for the area by adding to the areas of the squares enclosed partially by the curve.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: Graph 1: a. , ; b. . Graph 2: a. ; b. ,
. Graph 3: a. ; b. , .
59.
A unit circle is made up of n
wedges equivalent to the inner wedge in the figure. The base
of the inner triangle is 1 unit and its height is . The base of the outer triangle is
and the height is . Use this information to argue that the area of a unit circle is equal to π
.
Answer: Let A
be the area of the unit circle. The circle encloses n
congruent triangles each of area , so . Similarly, the circle is contained inside n
congruent triangles
each of area , so
. As , , so we conclude . Also, as , , so we also have . By the squeeze theorem for limits, we conclude that .
This file is copyright 2016, Rice University. All Rights Reserved.
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