Calculus for Business, Economics, Life Sciences and Social Sciences

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 7

Additional Integration Topics - all with Video Answers

Educators

Section 1

Area Between Curves

Problem 1

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals in Problems $1-4$ that represent the indicated shaded area.
Shaded area in Figure $\mathrm{B}$

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals in Problems $1-4$ that represent the indicated shaded area.
Shaded area in Figure $A$

Gregory Higby

Numerade Educator

Problem 3

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals in Problems $1-4$ that represent the indicated shaded area.
Shaded area in Figure $C$

Gregory Higby

Numerade Educator

Problem 4

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals in Problems $1-4$ that represent the indicated shaded area.
Shaded area in Figure $D$

Gregory Higby

Numerade Educator

Problem 5

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals.
Explain why $\int_{a}^{b} h(x) d x$ does not represent the area between the graph of $y=h(x)$ and the $x$ axis from $x=a$ to $x=b$ in Figure $C$.

Gregory Higby

Numerade Educator

Problem 6

Refer to Figures $A-D$ on page $418 .$ Set up definite integrals.
Explain why $\int_{a}^{b}[-h(x)] d x$ represents the area between the graph of $y=h(x)$ and the $x$ axis from $x=a$ to $x=b$ in Figure C.

Gregory Higby

Numerade Educator

Problem 7

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x+4 ; y=0 ; 0 \leq x \leq 4
$$

Narayan Hari

Numerade Educator

Problem 8

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x+10 ; y=0 ;-2 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 9

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{2}-20 ; y=0 ;-3 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 10

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{2}+2 ; y=0 ; 0 \leq x \leq 3
$$

Narayan Hari

Numerade Educator

Problem 11

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x^{2}+10 ; y=0 ;-3 \leq x \leq 3
$$

Narayan Hari

Numerade Educator

Problem 12

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-2 x^{2} ; y=0 ;-6 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 13

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x^{3}+1 ; y=0 ; 0 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 14

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x^{3}+3 ; y=0 ;-2 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 15

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=x(1-x) ; y=0 ;-1 \leq x \leq 0
$$

Narayan Hari

Numerade Educator

Problem 16

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-x(3-x) ; y=0 ; 1 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 17

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-e^{x} ; y=0 ;-1 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 18

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=e^{x} ; y=0 ; 0 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 19

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=\frac{1}{x}, y=0 ; 1 \leq x \leq e
$$

Narayan Hari

Numerade Educator

Problem 20

Find the area bounded by the graphs of the indicated equations over the given interval. Compute answers to three decimal places.
$$
y=-\frac{1}{x}, y=0 ;-1 \leq x \leq-\frac{1}{e}
$$

Narayan Hari

Numerade Educator

Problem 21

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=x ; y=-2 ; 0 \leq x \leq 3
$$

Gregory Higby

Numerade Educator

Problem 22

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=-x ; y=8 ; 0 \leq x \leq 5
$$

Gregory Higby

Numerade Educator

Problem 23

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=2 x+3 ; y=x-1 ; 4 \leq x \leq 6
$$

Gregory Higby

Numerade Educator

Problem 24

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=3 x+6 ; y=\frac{1}{2} x+1 ; 8 \leq x \leq 10
$$

Gregory Higby

Numerade Educator

Problem 25

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=-4 x+1 ; y=\frac{3}{2} x+1 ; 0 \leq x \leq 4
$$

Gregory Higby

Numerade Educator

Problem 26

Use a definite integral to find the area bounded by the graphs of the indicated equations over the given interval. Then check your answer by finding the area without using a definite integral.
$$
y=-2 x-3 ; y=x-3 ; 0 \leq x \leq 20
$$

Gregory Higby

Numerade Educator

Problem 27

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[a, b]$ in Figure $A$

Gregory Higby

Numerade Educator

Problem 28

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[c, d]$ in Figure $A$

Gregory Higby

Numerade Educator

Problem 29

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[b, d]$ in Figure $A$

Gregory Higby

Numerade Educator

Problem 30

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[a, c]$ in Figure $A$

Gregory Higby

Numerade Educator

Problem 31

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[c, d]$ in Figure $\mathrm{B}$

Gregory Higby

Numerade Educator

Problem 32

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[a, b]$ in Figure $B$

Gregory Higby

Numerade Educator

Problem 33

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[a, c]$ in Figure $\mathrm{B}$

Gregory Higby

Numerade Educator

Problem 34

Refer to Figures A and B. Set up definite integrals in Problems $27-34$ that re present the indicated shaded areas over the given intervals.
Over interval $[b, d]$ in Figure $\mathrm{B}$

Gregory Higby

Numerade Educator

Problem 35

Refer to Figures A and B. Set up definite integrals.
Referring to Figure B, explain how you would use definite integrals and the functions $f$ and $g$ to find the area bounded by the two functions from $x=a$ to $x=d$.

Gregory Higby

Numerade Educator

Problem 36

Refer to Figures A and B. Set up definite integrals.
Referring to Figure A, explain how you would use definite integrals to find the area between the graph of $y=f(x)$ and the $x$ axis from $x=a$ to $x=d$.

Gregory Higby

Numerade Educator

Problem 37

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x ; y=0 ;-2 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 38

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x+1 ; y=0 ;-1 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 39

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-4 ; y=0 ; 0 \leq x \leq 3
$$

Bobby Barnes

University of North Texas

Problem 40

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=4-x^{2} ; y=0 ; 0 \leq x \leq 4
$$

Bobby Barnes

University of North Texas

Problem 41

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-3 x ; y=0 ;-2 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 42

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=-x^{2}-2 x ; y=0 ;-2 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 43

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-3 x ; y=0 ;-2 \leq x \leq 2
$$

Bobby Barnes

University of North Texas

Problem 44

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=2 x+6 ; y=3 ;-1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 45

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=3 x^{2} ; y=12
$$

Gregory Higby

Numerade Educator

Problem 46

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
v=x^{2}: v=9
$$

Gregory Higby

Numerade Educator

Problem 47

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=4-x^{2} ; y=-5
$$

Bobby Barnes

University of North Texas

Problem 48

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-1 ; y=3
$$

Bobby Barnes

University of North Texas

Problem 49

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}+1 ; y=2 x-2 ;-1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 50

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{2}-1 ; y=x-2 ;-2 \leq x \leq 1
$$

Narayan Hari

Numerade Educator

Problem 51

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=e^{0.5 x} ; y=-\frac{1}{x}, 1 \leq x \leq 2
$$

Narayan Hari

Numerade Educator

Problem 52

Find the area bounded by the graphs of the indicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=\frac{1}{x}, y=-e^{x} ; 0.5 \leq x \leq 1
$$

Bobby Barnes

University of North Texas

Problem 53

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=\sqrt{9-x^{2} ;} y=0 ;-3 \leq x \leq 3
$$

Gregory Higby

Numerade Educator

Problem 54

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=\sqrt{25-x^{2}} ; y=0 ;-5 \leq x \leq 5
$$

Gregory Higby

Numerade Educator

Problem 55

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=-\sqrt{16-x^{2}} ; y=0 ; 0 \leq x \leq 4
$$

Gregory Higby

Numerade Educator

Problem 56

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=-\sqrt{36-x^{2}} ; y=0 ;-6 \leq x \leq 0
$$

Gregory Higby

Numerade Educator

Problem 57

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=-\sqrt{4-x^{2}} ; y=\sqrt{4-x^{2}} ;-2 \leq x \leq 2
$$

Gregory Higby

Numerade Educator

Problem 58

Set up a definite integral that represents the area bounded by the graphs of the indicated equations over the given interval. Find the areas to three decimal places.
$$
y=-\sqrt{100-x^{2}} ; y=\sqrt{100-x^{2}} ;-10 \leq x \leq 10
$$

Gregory Higby

Numerade Educator

Problem 59

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=3-5 x-2 x^{2} ; y=2 x^{2}+3 x-2
$$

Gregory Higby

Numerade Educator

Problem 60

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=3-2 x^{2} ; y=2 x^{2}-4 x
$$

Gregory Higby

Numerade Educator

Problem 61

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=-0.5 x+2.25 ; y=\frac{1}{x}
$$

Gregory Higby

Numerade Educator

Problem 62

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=x-4.25 ; y=-\frac{1}{x}
$$

Gregory Higby

Numerade Educator

Problem 63

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=e^{x} ; y=e^{-x} ; 0 \leq x \leq 4
$$

Gregory Higby

Numerade Educator

Problem 64

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=e^{x} ; y=-e^{-x} ; 1 \leq x \leq 2
$$

Gregory Higby

Numerade Educator

Problem 65

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{3} ; y=4 x
$$

Gregory Higby

Numerade Educator

Problem 66

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{3}+1 ; y=x+1
$$

Gregory Higby

Numerade Educator

Problem 67

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{3}-3 x^{2}-9 x+12 ; y=x+12
$$

Bobby Barnes

University of North Texas

Problem 68

Find the area bounded by the graphs of the in dicated equations over the given intervals (when stated). Compute answers to three decimal places.
$$
y=x^{3}-6 x^{2}+9 x ; y=x
$$

Bobby Barnes

University of North Texas

Problem 69

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=x^{3}-x^{2}+2 ; y=-x^{3}+8 x-2
$$

Bobby Barnes

University of North Texas

Problem 70

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=2 x^{3}+2 x^{2}-x ; y=-2 x^{3}-2 x^{2}+2 x
$$

Gregory Higby

Numerade Educator

Problem 71

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=e^{-x} ; y=3-2 x
$$

Bobby Barnes

University of North Texas

Problem 72

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=2-(x+1)^{2} ; y=e^{x+1}
$$

Bobby Barnes

University of North Texas

Problem 73

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=e^{x} ; y=5 x-x^{3}
$$

Bobby Barnes

University of North Texas

Problem 74

Use a graphing calculator to graph the equations and find relevant intersection points. Then find the area bounded by the curves. Compute answers to three decimal places.
$$
y=2-e^{x} ; y=x^{3}+3 x^{2}
$$

Bobby Barnes

University of North Texas

Problem 75

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{-x} ; y=\sqrt{\ln x ;} 2 \leq x \leq 5
$$

Wendi Zhao

Numerade Educator

Problem 76

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=x^{2}+3 x+1 ; y=e^{e^{x}} ;-3 \leq x \leq 0
$$

Wendi Zhao

Numerade Educator

Problem 77

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=e^{x^{2}} ; y=x+2
$$

Wendi Zhao

Numerade Educator

Problem 78

Use a numerical integration routine on a graphing calculator to find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places.
$$
y=\ln (\ln x) ; y=0.01 x
$$

Wendi Zhao

Numerade Educator

Problem 79

In the applications that follow, it is helpful to sketch graphs to get a clearer understanding of each problem and to inter pret results. A graphing calculator will prove use ful if you have one, but it is not necessary.
Using production and geological data, the management of an oil company estimates that oil will be pumped from a producing field at a rate given by $$ R(t)=\frac{100}{t+10}+10 \quad 0 \leq t \leq 15 $$. where $R(t)$ is the rate of production (in thousands of barrels per year) $t$ years after pumping begins. Find the area between the graph of $R$ and the $t$ axis over the interval [5,10] and interpret the results.

Bobby Barnes

University of North Texas

Problem 80

If the rate is found to be $$ R(t)=\frac{100 t}{t^{2}+25}+4 \quad 0 \leq t \leq 25 $$ then find the area between the graph of $R$ and the $t$ axis over the interval [5,15] and interpret the results.

Bobby Barnes

University of North Texas

Problem 81

An amusement company maintains records for each video game it installs in an arcade. Suppose that $C(t)$ and $R(t)$ represent the total accumulated costs and revenues (in thousands of dollars), respectively, $t$ years after a particular game has been installed. If $$ C^{\prime}(t)=2 \quad \text { and } \quad R^{\prime}(t)=9 e^{-0.3 t} $$ then find the area between the graphs of $C^{\prime}$ and $R^{\prime}$ over the interval on the $t$ axis from 0 to the useful life of the game and interpret the results.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 82

Repeat Problem 81 if $$ C^{\prime}(t)=2 t \quad \text { and } \quad R^{\prime}(t)=5 t e^{-0.1 t^{2}} $$

Monica Miller

Numerade Educator

Problem 83

In a study on the effects of World War II on the U.S. economy, an economist used data from the U.S. Census Bureau to produce the following Lorenz curves for the distribution of U.S. income in 1935 and in 1947: $$f(x)=x^{2.4}$$ $$g(x)=x^{1.6}$$ Find the Gini index of income concentration for each Lorenz curve and interpret the results.

Bobby Barnes

University of North Texas

Problem 84

Using data from the U.S. Census Bureau, an economist produced the following Lorenz curves for the distribution of U.S. income in 1962 and in 1972 : $$f(x)=\frac{3}{10} x+\frac{7}{10} x^{2} \quad$$ $$g(x)=\frac{1}{2} x+\frac{1}{2} x^{2} \quad$$ Find the Gini index of income concentration for each Lorenz curve and interpret the results.

Maria Dearborn

Numerade Educator

Problem 85

Lorenz curves also can provide a relative measure of the distribution of a country's total assets. Using data in a report by the U.S. Congressional Joint Economic Committee, an economist produced the following Lorenz curves for the distribution of total U.S. assets in 1963 and in 1983: $$ \begin{array}{l} f(x)=x^{10} \quad\\ g(x)=x^{12} \quad \end{array} $$ Find the Gini index of income concentration for each Lorenz curve and interpret the results.

Maria Dearborn

Numerade Educator

Problem 86

The government of a small country is planning sweeping changes in the tax structure in order to provide a more equitable distribution of income. The Lorenz curves for the current income distribution and for the projected income distribution after enactment of the tax changes are as follows: $$f(x)=x^{2.3}$$ $$g(x)=0.4 x+0.6 x^{2}$$ Find the Gini index of income concentration for each Lorenz curve. Will the proposed changes provide a more equitable income distribution? Explain

Bobby Barnes

University of North Texas

Problem 87

The data in the following table describe the distribution of wealth in a country:
(A) Use quadratic regression to find the equation of a Lorenz curve for the data.
(B) Use the regression equation and a numerical integration routine to approximate the Gini index of income

Bobby Barnes

University of North Texas

Problem 88

Refer to Problem $87 .$
(A) Use cubic regression to find the equation of a Lorenz curve for the data.
(B) Use the cubic regression equation you found in Part (A) and a numerical integration routine to approximate the Gini index of income concentration.

Angela Guo

Numerade Educator

Problem 89

A yeast culture is growing at a rate of $W^{\prime}(t)=0.3 e^{0.1 t}$ grams per hour. Find the area between the graph of $W^{\prime}$ and the $t$ axis over the interval [0,10] and interpret the results.

Bobby Barnes

University of North Texas

Problem 90

The instantaneous rate of change in demand for U.S. lumber since $1970(t=0),$ in billions of cubic feet per year, is given by $$ Q^{\prime}(t)=12+0.006 t^{2} \quad 0 \leq t \leq 50 $$ Find the area between the graph of $Q^{\prime}$ and the $t$ axis over the interval $[15,20],$ and interpret the results.

Bobby Barnes

University of North Texas

Problem 91

A college language class was chosen for a learning experiment. Using a list of 50 words, the experiment measured the rate of vocabulary memorization at different times during a continuous 5 -hour study session. The average rate of learning for the entire class was inversely proportional to the time spent studying and was given approximately by
$$ V^{\prime}(t)=\frac{15}{t} \quad 1 \leq t \leq 5 $$ Find the area between the graph of $V^{\prime}$ and the $t$ axis over the interval $[2,4],$ and interpret the results.

Priyanka Sadarangani

Priyanka Sadarangani

Numerade Educator

Problem 92

Repeat Problem 91 if $V^{\prime}(t)=13 / t^{1 / 2}$ and the interval is changed to [1,4].

Tyler Moulton

Numerade Educator