Calculus Volume 2

Gilbert Strang,

Chapter 1

Integration - all with Video Answers

Educators

+ 1 more educators

Section 1

Approximating Areas

Problem 1

State whether the given sums are equal or unequal.
$$
\begin{array}{l}{\text { a. } \sum_{i=1}^{10} i \text { and } \sum_{k=1}^{10} k} \\ {\text { b. } \sum_{i=1}^{10} i \text { and } \sum_{i=6}^{15}(i-5)} \\ {\text { c. } \sum_{i=1}^{10} i(i-1) \text { and } \sum_{j=0}^{9}(j+1) j} \\ {\text { d. } \sum_{i=1}^{10} i(i-1) \text { and } \sum_{j=0}^{10}\left(k^{2}-k\right)}\end{array}
$$

Taylor Shimono

Taylor Shimono

Numerade Educator

Problem 2

In the following exercises, use the rules for sums of powers of integers to compute the sums.
$$
\sum_{i=5}^{10} i
$$

Amrita Bhasin

Numerade Educator

Problem 3

In the following exercises, use the rules for sums of powers of integers to compute the sums.
$$
\sum_{i=5}^{10} i^{2}
$$

Taylor Shimono

Numerade Educator

Problem 4

Suppose that $\sum_{i=1}^{100} a_{i}=15$ and $\sum_{i=1}^{100} b_{i}=-12 .$ In the following exercises, compute the sums.
$$
\sum_{i=1}^{100}\left(a_{i}+b_{i}\right)
$$

Amrita Bhasin

Numerade Educator

Problem 5

Suppose that $\sum_{i=1}^{100} a_{i}=15$ and $\sum_{i=1}^{100} b_{i}=-12 .$ In the following exercises, compute the sums.
$$
\sum_{i=1}^{100}\left(a_{i}-b_{i}\right)
$$

Taylor Shimono

Numerade Educator

Problem 6

Suppose that $\sum_{i=1}^{100} a_{i}=15$ and $\sum_{i=1}^{100} b_{i}=-12 .$ In the following exercises, compute the sums.
$$
\sum_{i=1}^{100}\left(3 a_{i}-4 b_{i}\right)
$$

Amrita Bhasin

Numerade Educator

Problem 7

Suppose that $\sum_{i=1}^{100} a_{i}=15$ and $\sum_{i=1}^{100} b_{i}=-12 .$ In the following exercises, compute the sums.
$$
\sum_{i=1}^{100}\left(5 a_{i}+4 b_{i}\right)
$$

Taylor Shimono

Numerade Educator

Problem 8

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$$
\sum_{k=1}^{20} 100\left(k^{2}-5 k+1\right)
$$

Amrita Bhasin

Numerade Educator

Problem 9

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$$
\sum_{j=1}^{50}\left(j^{2}-2 j\right)
$$

Taylor Shimono

Numerade Educator

Problem 10

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$$
\sum_{j=11}^{20}\left(j^{2}-10 j\right)
$$

Amrita Bhasin

Numerade Educator

Problem 11

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.
$$
\sum_{k=1}^{25}\left[(2 k)^{2}-100 k\right]
$$

Taylor Shimono

Numerade Educator

Problem 12

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
L_{4} \text { for } f(x)=\frac{1}{x-1} \text { on }[2,3]
$$

Amrita Bhasin

Numerade Educator

Problem 13

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
R_{4} \text { for } g(x)=\cos (\pi x) \text { on }[0,1]
$$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 14

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
L_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 15

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
R_{6} \text { for } f(x)=\frac{1}{x(x-1)} \text { on }[2,5]
$$

Taylor Shimono

Numerade Educator

Problem 16

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
R_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2]
$$

Amrita Bhasin

Numerade Educator

Problem 17

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
L_{4} \text { for } \frac{1}{x^{2}+1} \text { on }[-2,2]
$$

Taylor Shimono

Numerade Educator

Problem 18

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
R_{4} \text { for } x^{2}-2 x+1 \text { on }[0,2]
$$

Amrita Bhasin

Numerade Educator

Problem 19

Let $L_{n}$ denote the left-endpoint sum using $n$ subintervals and let $R_{n}$ 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.
$$
L_{8} \text { for } x^{2}-2 x+1 \text { on }[0,2]
$$

Taylor Shimono

Numerade Educator

Problem 20

Compute the left and right Riemann sums- $L_{4}$ and $R_{4}$ respectively- for $f(x)=(2-|x|)$ on $[-2,2] .$ Compute their average value and compare it with the area under the graph of $f .$

Amrita Bhasin

Numerade Educator

Problem 21

Compute the left and right Riemann sums- $-L_{6}$ and $R_{6}$ , respectively- for $f(x)=(3-|3-x|) \quad$ on $\quad[0,6]$ Compute their average value and compare it with the area under the graph of $f .$

Taylor Shimono

Numerade Educator

Problem 22

Compute the left and right Riemann sums $-L_{4}$ and $R_{4},$ respectively- for $f(x)=\sqrt{4-x^{2}}$ on $[-2,2]$ and compare their values.

Amrita Bhasin

Numerade Educator

Problem 23

Compute the left and right Riemann sums $-L_{6}$ and $R_{6}$ , respectively- for $f(x)=\sqrt{9-(x-3)^{2}}$ on $[0,6]$ and compare their values.

Taylor Shimono

Numerade Educator

Problem 24

Express the following endpoint sums in sigma notation but do not evaluate them.
$$
L_{30} \text { for } f(x)=x^{2} \text { on }[1,2]
$$

Amrita Bhasin

Numerade Educator

Problem 25

Express the following endpoint sums in sigma notation but do not evaluate them.
$$
L_{10} \text { for } f(x)=\sqrt{4-x^{2}} \text { on }[-2,2]
$$

Taylor Shimono

Numerade Educator

Problem 26

Express the following endpoint sums in sigma notation but do not evaluate them.
$$
R_{20} \text { for } f(x)=\sin x \text { on }[0, \pi]
$$

Bobby Barnes

University of North Texas

Problem 27

Express the following endpoint sums in sigma notation but do not evaluate them.
$$
R_{100} \text { for } \ln x \text { on }[1, e]
$$

Taylor Shimono

Numerade Educator

Problem 28

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 between the left and right endpoint sums?
$$
[\mathrm{T}] L_{100} \text { and } R_{100} \text { for } y=x^{2} \text { on the interval }[0,1]
$$

Amrita Bhasin

Numerade Educator

Problem 29

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 between the left and right endpoint sums?
$$
L_{100} \text { and } R_{100} \text { for } y=x^{2} \text { on the interval }[0,1]
$$

Taylor Shimono

Numerade Educator

Problem 30

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 between the left and right endpoint sums?
$$
[\mathrm{T}] L_{50} \text { and } R_{50} \text { for } y=\frac{x+1}{x^{2}-1} \text { on the interval }[2,4]
$$

Amrita Bhasin

Numerade Educator

Problem 31

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 between the left and right endpoint sums?
$$
[\mathrm{T}] L_{100} \text { and } R_{100} \text { for } y=x^{3} \text { on the interval }[-1,1]
$$

Taylor Shimono

Numerade Educator

Problem 32

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 between the left and right endpoint sums?
$$
[\mathrm{T}] L_{50} \text { and } R_{50} \text { for } y=\tan (x) \text { on the interval }\left[0, \frac{\pi}{4}\right]
$$

Amrita Bhasin

Numerade Educator

Problem 33

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 between the left and right endpoint sums?
$$
[\mathrm{T}] L_{100} \text { and } R_{100} \text { for } y=e^{2 x} \text { on the interval }[-1,1]
$$

Taylor Shimono

Numerade Educator

Problem 34

Let $t_{j}$ denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014 . If there were a total of 21 stages, interpret $\sum_{j=1}^{21} t_{j}$

Amrita Bhasin

Numerade Educator

Problem 35

Let $r_{j}$ denote the total rainfall in Portland on the jth day of the year in 2009. Interpret $\sum_{j=1}^{31} r_{j}$

Taylor Shimono

Numerade Educator

Problem 36

Let $d_{j}$ denote the hours of daylight and $\delta_{j}$ denote the increase in the hours of daylight from day $j-1$ to day $j$ in Fargo, North Dakota, on the jth day of the year. Interpret $d_{1}+\sum_{j=2}^{365} \delta_{j}$

Amrita Bhasin

Numerade Educator

Problem 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 $\frac{1}{10} \mathrm{mi}$ to his daily routine each week, what is the total mileage on Joe's shoes after 25 weeks?

Taylor Shimono

Numerade Educator

Problem 38

The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide $\left(\mathrm{CO}_{2}\right)$ each decade since $1960,$ in parts per million
(ppm). Estimate the total increase in atmospheric $\mathrm{CO}_{2}$ between 1964 and 2013 .

Amrita Bhasin

Numerade Educator

Problem 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 .

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 40

The following table gives the approximate increase in dollars in the average price of a gallon of gas per decade since 1950 . If the average price of a gallon of gas in 2010 was $\$ 2.60$ , what was the average price of a gallon of gas in 1950$?$

Amrita Bhasin

Numerade Educator

Problem 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 .

Taylor Shimono

Numerade Educator

Problem 42

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

Amrita Bhasin

Numerade Educator

Problem 43

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

Taylor Shimono

Numerade Educator

Problem 44

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

Amrita Bhasin

Numerade Educator

Problem 45

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $L_{8} .$

Taylor Shimono

Numerade Educator

Problem 46

[T] Use a computer algebra system to compute the Riemann sum, $\quad L_{N}, \quad$ for $\quad N=10,30,50$ for $f(x)=\sqrt{1-x^{2}}$ on $[-1,1]$

Amrita Bhasin

Numerade Educator

Problem 47

[T] Use a computer algebra system to compute the Riemann sum, $L_{N}, \quad$ for $\quad N=10,30,50$ for $f(x)=\frac{1}{\sqrt{1+x^{2}}}$ on $[-1,1]$

Taylor Shimono

Numerade Educator

Problem 48

[T] Use a computer algebra system to compute the Riemann sum, $L_{N},$ for $N=10,30,50$ for $f(x)=\sin ^{2} x$ on $[0,2 \pi]$ . Compare these estimates with $\pi$

Amrita Bhasin

Numerade Educator

Problem 49

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100 .$ How do these estimates compare with the exact answers, which you can find via geometry?
$$
[\mathrm{T}] \mathrm{y}=\cos (\pi x) \text { on the interval }[0,1]
$$

Taylor Shimono

Numerade Educator

Problem 50

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100 .$ How do these estimates compare with the exact answers, which you can find via geometry?
$$
[\mathrm{T}] \mathrm{y}=3 x+2 \text { on the interval }[3,5]
$$

Amrita Bhasin

Numerade Educator

Problem 51

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
$$
\begin{array}{l}{\text { [T] } y=x^{4}-5 x^{2}+4 \text { on the interval }[-2,2],} \\ {\text { which has an exact area of } \frac{32}{15}}\end{array}
$$

Taylor Shimono

Numerade Educator

Problem 52

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
$$
\begin{array}{l}{\text { [T] } y=\ln x \text { on the interval }[1,2], \text { which has an }} \\ {\text { exact area of } 2 \ln (2)-1}\end{array}
$$

Amrita Bhasin

Numerade Educator

Problem 53

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
Explain why, if $f(a) \geq 0$ and $f$ is increasing on $[a, b],$ that the left endpoint estimate is a lower bound for the area below the graph of $f$ on $[a, b]$

Taylor Shimono

Numerade Educator

Problem 54

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
Explain why, if $f(b) \geq 0$ and $f$ is decreasing on $[a, b],$ that the left endpoint estimate is an upper bound for the area below the graph of $f$ on $[a, b]$

Amrita Bhasin

Numerade Educator

Problem 55

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
Show that, in general,
$$
R_{N}-L_{N}=(b-a) \times \frac{f(b)-f(a)}{N}
$$

Taylor Shimono

Numerade Educator

Problem 56

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
Explain why, if $f$ is increasing on $[a, b],$ the error between either $L_{N}$ or $R_{N}$ and the area $A$ below the graph of $f$ is at most $(b-a) \frac{f(b)-f(a)}{N}$

Amrita Bhasin

Numerade Educator

Problem 57

In the following exercises, use a calculator or a computer program to evaluate the endpoint sums $R_{N}$ and $L_{N}$ for $N=1,10,100$ .
For each of the three graphs:
a. Obtain a lower bound $L(A)$ for the area enclosed by the curve by adding the areas of the squares
enclosed completely by the curve.
b. Obtain an upper bound $U(A)$ for the area by adding to $L(A)$ the areas $B(A)$ of the squares
enclosed partially by the curve.

Taylor Shimono

Numerade Educator

Problem 58

In the previous exercise, explain why $L(A)$ gets no smaller while $U(A)$ gets no larger as the squares are subdivided into four boxes of equal area.

Amrita Bhasin

Numerade Educator

Problem 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 $\sin \left(\frac{\pi}{n}\right) .$ The base of the outer triangle
is $\quad B=\cos \left(\frac{\pi}{n}\right)+\sin \left(\frac{\pi}{n}\right) \tan \left(\frac{\pi}{n}\right)$ and the height is $H=B \sin \left(\frac{2 \pi}{n}\right)$ . Use this information to argue that the area
of a unit circle is equal to $\pi .$

Taylor Shimono

Numerade Educator