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Calculus

Laura Taalman, Peter Kohn

Chapter 4

Definite Integrals - all with Video Answers

Educators

AW

Section 1

Definite Integrals

00:54

Problem 1

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k=1$
(b) True or False: $\sum_{k=0}^{n} \frac{1}{k+1}+\sum_{k=1}^{n} k^{2}$ is equal to $\sum_{k=0}^{n} \frac{k^{3}+k^{2}+1}{k+1}$
(c) True or False: $\sum_{k=1}^{n} \frac{1}{k+1}+\sum_{k=0}^{n} k^{2}$ is equal to $\sum_{k=1}^{n} \frac{k^{3}+k^{2}+1}{k+1}$
(d) True or False: $\left(\sum_{k=1}^{n} \frac{1}{k+1}\right)\left(\sum_{k=1}^{n} k^{2}\right)$ is equal to $\sum_{k=1}^{n} \frac{k^{2}}{k+1}$

Amy Jiang
Amy Jiang
Numerade Educator
04:12

Problem 2

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A sum that would not be suitable for expressing in sigma notation.
(b) Two different sigma notation expressions of the same sum.
(c) A sum from $k=1$ to $k=n$ that is smaller for $n=10$ than it is for $n=5$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
01:41

Problem 3

Explain why it would be difficult to write the sum $\frac{1}{3}+\frac{1}{4}+$ $\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{12}+\frac{1}{13}$ in sigma notation.

Samriddhi Singh
Samriddhi Singh
Numerade Educator
05:01

Problem 4

Use a sentence to describe what the notation $\sum_{k=2}^{100} \sqrt{k}$ means. (Hint: Start with "The sum of...")

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:25

Problem 5

Use a sentence to describe what the notation $\sum_{k=3}^{87} k^{2}$ means. (Hint: Start with "The sum of...")

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:23

Problem 6

Consider the general sigma notation $\sum_{k=m}^{n} a_{k} .$ What do we mean when we say that $a_{k}$ is a function of $k ?$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
11:39

Problem 7

Consider the sum $\sum_{i=p}^{q} b_{i}$
(a) Write out this sum in expanded form (i.e., without sigma notation).
(b) What is the index of the sum? What is the starting value? What is the ending value? Which part of the notation describes the form of each of the terms in the sum?
(c) Do $p$ and $q$ have to be integers? Can they be negative? What about $b_{i} ?$ What else can you say about $p$ and $q$ ?

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:15

Problem 8

Consider the sum $\sum_{k=2}^{5} \frac{k}{1-k}$. Identify the terms $a_{2}, a_{3}$ $a_{4},$ and $a_{5}$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:40

Problem 9

Consider the sum $\sum_{k=m}^{n} a_{k}=9+16+25+36+49$
What is $a_{k} ?$ What is $m$ ? What is $n$ ?

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:37

Problem 10

Show that $\sum_{k=3}^{9} \frac{k}{k+1}$ is equal to $\sum_{k=4}^{10} \frac{k-1}{k}$ by writing out the terms in each sum.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
09:49

Problem 11

Show that $\sum_{k=0}^{8} \frac{1}{k^{2}+1}$ is equal to $2 \sum_{k=0}^{8} \frac{1}{2 k^{2}+2}$ by writing out the terms in each sum.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
08:19

Problem 12

Write the sum $\frac{4}{7}+\frac{5}{8}+\frac{6}{9}+\frac{7}{10}+\frac{8}{11}$ in sigma notation in three ways: with a starting value of (a) $k=4$
(b) $k=7$, and (c) $k=5$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:13

Problem 13

Write the sum $2+\frac{2}{4}+\frac{2}{9}+\frac{2}{16}+\frac{2}{25}$ in sigma notation in three ways: with a starting value of
(a) $k=1$,
(b) $k=2$ and (c) $k=0$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:01

Problem 14

Split the sum $\sum_{k=4}^{11} \sqrt{k}$ into three sums, each in sigma notation, where the first sum has two terms and the last two sums each have three terms.

Samriddhi Singh
Samriddhi Singh
Numerade Educator
09:03

Problem 15

Verify that $\sum_{k=1}^{n} k$ is equal to $\frac{n(n+1)}{2}$ for the cases (a) $n=2,$ (b) $n=8,$ and (c) $n=9 .$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
10:33

Problem 16

Verify that $\sum_{k=1}^{n} k^{2}$ is equal to $\frac{n(n+1)(2 n+1)}{6}$ for the cases
(a) $n=1$,
(b) $n=5,$ and
(c) $n=10$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:21

Problem 17

State algebraic formulas that express the following sums, where $n$ is a positive integer:
(a) $\sum_{k=1}^{n} 1$
(b) $\sum_{k=1}^{n} k$
(c) $\sum_{k=1}^{n} k^{2}$
(d) $\sum_{k=1}^{n} k^{3}$

AW
Abdul Wasay
Numerade Educator
08:40

Problem 18

Explain why terms in the sum in Example 6 with $n=4$ are completely different from the terms in the sum when $n=3 .$ How can the sum from $k=1$ to $k=4$ be smaller than the sum from $k=1$ to $k=3$ ? What will happen as $n$ gets larger in this example?

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:20

Problem 19

Considering the discussion at the end of the stoplight example in the reading, would you expect that the area under the graph of a function $f$ is related to the derivative $f^{\prime} ?$ Or would you expect that the area under the graph of a derivative function $f^{\prime}$ is related to the function $f ?$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
01:29

Problem 20

Consider again the stoplight example from the reading. In making an approximation for distance travelled, why do we assume that velocity is constant on small subintervals? What are some different ways that we could choose which velocity to use on each subinterval? Illustrate a couple of these ways with graphs that involve rectangles.

Samriddhi Singh
Samriddhi Singh
Numerade Educator
03:49

Problem 21

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
3+3+3+3+3+3+3+3
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:39

Problem 22

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
\frac{4}{3}+\frac{5}{4}+\frac{6}{5}+\frac{7}{6}+\frac{8}{7}+\frac{9}{8}+\frac{10}{9}+\frac{11}{10}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:38

Problem 23

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
3+\frac{4}{8}+\frac{5}{27}+\frac{6}{64}+\frac{7}{125}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:51

Problem 24

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+\frac{1}{49}+\frac{1}{64}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:08

Problem 25

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
5+10+17+26+37+50+65+82+101
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:56

Problem 26

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
9+12+15+18+21+24+27
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:45

Problem 27

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\cdots+\frac{n}{n}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:02

Problem 28

Write each of the sums in sigma notation. Identify $m, n,$ and $a_{k}$ in each problem.
$$
-2^{n}-1^{n}+0^{n}+1^{n}+\cdots+n^{n}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:06

Problem 29

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=4}^{9} k^{2}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:05

Problem 30

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=0}^{6} \frac{2}{k+1}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:55

Problem 31

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=0}^{5}\left(\frac{1}{2} k\right)^{2}\left(\frac{1}{2}\right)
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:40

Problem 32

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=3}^{10} \ln k
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:15

Problem 33

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=0}^{9} \sqrt{3+\frac{1}{10} k}\left(\frac{1}{10}\right)
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:21

Problem 34

Write out each sum in expanded form, and then calculate the value of the sum.
$$
\sum_{k=1}^{4}\left((2+k)^{2}+1\right)
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:57

Problem 35

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=1}^{n}(3-k)
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:37

Problem 36

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=1}^{n}\left(k^{3}-10 k^{2}+2\right)
$$

AW
Abdul Wasay
Numerade Educator
06:11

Problem 37

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=3}^{n}(k+1)^{2}
$$

AW
Abdul Wasay
Numerade Educator
03:07

Problem 38

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=1}^{n} \frac{k^{3}-1}{4}
$$

AW
Abdul Wasay
Numerade Educator
03:31

Problem 39

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=1}^{n} \frac{k^{3}-1}{n^{4}}
$$

Samriddhi Singh
Samriddhi Singh
Numerade Educator
04:54

Problem 40

Find a formula for each of the sums and then use these formulas to calculate each sum for $n=100$, $n=500,$ and $n=1000 .$
$$
\sum_{k=1}^{n} \frac{k^{2}+k+1}{n^{3}}
$$

Samriddhi Singh
Samriddhi Singh
Numerade Educator
05:33

Problem 41

Write each expression in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).
$$
2 \sum_{k=0}^{100} a_{k}-\sum_{k=3}^{101} a_{k}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:33

Problem 42

Write each expression in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).
$$
\sum_{k=1}^{40} \frac{1}{k}-\sum_{k=0}^{39} \frac{1}{k+1}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
07:05

Problem 43

Write each expression in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).
$$
3 \sum_{k=2}^{25} k^{2}+2 \sum_{k=2}^{24} k-\sum_{k=0}^{25} 1
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:27

Problem 44

Find the sum or quantity without completely expanding or calculating any sums.
Given $\sum_{k=3}^{10} a_{k}=12$ and $\sum_{k=2}^{10} a_{k}=23,$ find $a_{2}$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:19

Problem 45

Find the sum or quantity without completely expanding or calculating any sums.
Given $\sum_{k=1}^{4} a_{k}=7, \sum_{k=0}^{4} b_{k}=10,$ and $a_{0}=2,$ find the value of $\sum_{k=0}^{4}\left(2 a_{k}+3 b_{k}\right)$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:29

Problem 46

Find the sum or quantity without completely expanding or calculating any sums.
Given $\sum_{k=0}^{25} k=325$ and $\sum_{k=3}^{28}(k-3)^{2}=14,910,$ find the value of $\sum_{k=3}^{25}\left(k^{2}-5 k+9\right)$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
07:04

Problem 47

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{k^{2}+k+1}{n^{3}}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:28

Problem 48

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(k^{2}+k+1\right)
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:45

Problem 49

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{(k+1)^{2}}{n^{3}-1}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:59

Problem 50

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{k^{2}+k+1}{n^{2}}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
06:02

Problem 51

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(1+\frac{k}{n}\right)^{2} \cdot \frac{1}{n}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:22

Problem 52

Determine which of the limit of sums are infinite and which are finite, For each limit of sums that is finite, compute its value.
$$
\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{k^{3}}{n^{4}+n+1}
$$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
05:54

Problem 53

Considering the stoplight example in the reading with velocity $v(t)=-0.22 t^{2}+8.8 t$ as shown next at the left, approximate the distance travelled by dividing the time interval [0,40] into eight pieces and assuming constant velocity on each piece. Interpret this distance in terms of rectangles on the graph of $v(t)$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
01:42

Problem 54

Suppose you drive on a racetrack for 10 minutes with velocity as shown in the graph at the right.
(a) Describe in words the behavior of your race car over the 10 minutes as shown in the graph.
(b) Find a piecewise-defined formula for your velocity $v(t),$ in miles per hour, $t$ hours after you start from rest. (Note that 1 minute is $\frac{1}{60}$ of an hour.)
(c) Approximate the distance you travelled over the 10 minutes by using 10 subintervals of 1 minute over which you assume a constant velocity. Illustrate this approximation by showing rectangles on the graph of $v(t)$
(d) Given that distance travelled is the area under the velocity graph, use triangles and squares to calculate the exact distance travelled.

AG
Ankit Gupta
Numerade Educator
07:03

Problem 55

The table that follows describes the activity in a college tuition savings account over four years. Notice that 2008 was a particularly bad year for investing! Let $I(t)$ be the amount by which your account increased or decreased in year $t,$ and let $B(t)$ be the balance of your account at the end of year $t$
$$
\begin{array}{|r|c|c|c|c|}
\hline \text { Year } & 2005 & 2006 & 2007 & 2008 \\
\hline \text { Deposited } & \$ 600 & \$ 1200 & \$ 1200 & \$ 1200 \\
\hline \text { Earnings } & \$ 10 & \$ 183 & \$ 317 & \$-1650 \\
\hline \text { Increase } & \$ 610 & \$ 1383 & \$ 1517 & \$-450 \\
\hline \text { Balance } & \$ 610 & \$ 1994 & \$ 3512 & \$ 3061 \\
\hline
\end{array}
$$
(a) Describe in your own words how $B(t)$ is the accumulation function for $I(t)$.
(b) Plot a step-function graph of $I(t),$ and describe how $B(t)$ relates to the area under that graph.
(c) What, if anything, can you say about $B(t)$ when $I(t)$ is positive? Negative? If you had to guess that one of these functions was related to the derivative of the other, which one would it be?

Muhammad Saleem
Muhammad Saleem
Numerade Educator
01:07

Problem 56

Suppose $100 \mathrm{mg}$ of a drug is administered to a patient each morning in pill form and it is known that after 24 hours the body processes $80 \%$ of the drug from such a pill, leaving $20 \%$ of the drug in the body. The amount of the drug in the body right after the first pill is taken is $A(1)=100 \mathrm{mg} .24$ hours later, after the second pill has been taken, the amount in the body is $A(2)=100(0.2)+$ $100=120 \mathrm{mg} .48$ hours later, the amount in the body right after taking the third pill is $A(3)=100(0.2)(0.2)+$ $100(0.2)+100=124 \mathrm{mg}$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
10:35

Problem 57

Suppose the government enacts a $\$ 10$ billion tax cut and that the people who save money from this tax cut will spend $70 \%$ of it and save the rest. This generates $\$ 10(0.7)=\$ 7$ billion of extra income for other people. Assume these people also spend $70 \%$ of their extra income, and that these transactions continue. Let $A(n)$ be the accumulated amount of spending, in billions of dollars, that has occurred after $n$ such transactions. For example, $A(1)$ is the amount of spending that has occurred after the first group of people has spent its money, so $A(1)=$ $\$ 10(0.7)=\$ 7$ billion. $A(2)$ is the amount of spending that has occurred after the first and second groups of people have spent their money, so $A(2)=\$ 10(0.7)+\$ 7(0.7)=$ $\$ 11.9$ billion, as shown in the following graph:

Muhammad Saleem
Muhammad Saleem
Numerade Educator
04:32

Problem 58

Give a simple proof that $\sum_{k=5}^{n}\left(a_{k}+b_{k}\right)=\sum_{k=5}^{n} a_{k}+$ $\sum_{k=5}^{n} b_{k}$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:54

Problem 59

Give a simple proof that $\sum_{k=0}^{n} 3 a_{k}=3 \sum_{k=0}^{n} a_{k}$

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:16

Problem 60

Give a simple proof that if $n$ is a positive integer and $c$ is any real number, then $\sum_{k=1}^{n} c=c n$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:38

Problem 61

Prove part (b) of Theorem 4.4 in the case when $n$ is even: If $n$ is a positive even integer, then $\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$. (Hint: Try some examples first, such as $n=6$ and $n=10,$ and think about how to group the terms to get the sum quickly.)

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:17

Problem 62

Prove part (b) of Theorem 4.4 in the case when $n$ is odd: If $n$ is a positive odd integer, then $\sum_{k=1}^{n} k=\frac{n(n+1)}{2}$. (Hint:
Use a method similar to the one for the previous exercise, but take note of what happens with the extra middle term of the sum.

Samriddhi Singh
Samriddhi Singh
Numerade Educator