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Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

Chapter 6

The Definite Integral - all with Video Answers

Educators

Section 1

Antidifferentiation

00:44

Problem 1

Find all antiderivatives of each following function:
$$f(x)=x$$

Linh Vu
Linh Vu
Numerade Educator
01:08

Problem 2

Find all antiderivatives of each following function:
$$f(x)=9 x^{8}$$

Linh Vu
Linh Vu
Numerade Educator
00:36

Problem 3

Find all antiderivatives of each following function:
$$f(x)=e^{3 x}$$

Linh Vu
Linh Vu
Numerade Educator
00:34

Problem 4

Find all antiderivatives of each following function:
$$f(x)=e^{-3 x}$$

Linh Vu
Linh Vu
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00:26

Problem 5

Find all antiderivatives of each following function:
$$f(x)=3$$

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 6

Find all antiderivatives of each following function:
$$f(x)=-4 x$$

Linh Vu
Linh Vu
Numerade Educator
01:05

Problem 7

Determine the following:
$$\int 4 x^{3} d x$$

Linh Vu
Linh Vu
Numerade Educator
01:05

Problem 8

Determine the following:
$$\int \frac{x}{3} d x$$

Linh Vu
Linh Vu
Numerade Educator
00:25

Problem 9

Determine the following:
$$\int 7 d x$$

Linh Vu
Linh Vu
Numerade Educator
00:36

Problem 10

Determine the following:
$\int k^{2} d x(k$ a constant $)$

Linh Vu
Linh Vu
Numerade Educator
01:14

Problem 11

Determine the following:
$\int \frac{x}{c} d x(c$ a constant $\neq 0)$

Linh Vu
Linh Vu
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00:47

Problem 12

Determine the following:
$$\int x \cdot x^{2} d x$$

Linh Vu
Linh Vu
Numerade Educator
01:56

Problem 13

Determine the following:
$$\int\left(\frac{2}{x}+\frac{x}{2}\right) d x$$

Linh Vu
Linh Vu
Numerade Educator
00:48

Problem 14

Determine the following:
$$\int \frac{1}{7 x} d x$$

Linh Vu
Linh Vu
Numerade Educator
00:54

Problem 15

Determine the following:
$$\int x \sqrt{x} d x$$

Linh Vu
Linh Vu
Numerade Educator
02:25

Problem 16

Determine the following:
$$\int\left(\frac{2}{\sqrt{x}}+2 \sqrt{x}\right) d x$$

Linh Vu
Linh Vu
Numerade Educator
01:33

Problem 17

Determine the following:
$$\int\left(x-2 x^{2}+\frac{1}{3 x}\right) d x$$

Linh Vu
Linh Vu
Numerade Educator
01:54

Problem 18

Determine the following:
$$\int\left(\frac{7}{2 x^{3}}-\sqrt[3]{x}\right) d x$$

Linh Vu
Linh Vu
Numerade Educator
00:54

Problem 19

Determine the following:
$$\int 3 e^{-2 x} d x$$

Linh Vu
Linh Vu
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00:46

Problem 20

Determine the following:
$$\int e^{-x} d x$$

Linh Vu
Linh Vu
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00:31

Problem 21

Determine the following:
$$\int e d x$$

Linh Vu
Linh Vu
Numerade Educator
00:56

Problem 22

Determine the following:
$$\int \frac{7}{2 e^{2 x}} d x$$

Linh Vu
Linh Vu
Numerade Educator
01:31

Problem 23

Determine the following:
$$\int-2\left(e^{2 x}+1\right) d x$$

Linh Vu
Linh Vu
Numerade Educator
01:51

Problem 24

Determine the following:
$$\int\left(-3 e^{-x}+2 x-\frac{e^{.5 x}}{2}\right) d x$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:14

Problem 25

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int 5 e^{-2 t} d t=k e^{-2 t}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:12

Problem 26

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int 3 e^{t / 10} d t=k e^{t / 10}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:50

Problem 27

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int 2 e^{4 x-1} d x=k e^{4 x-1}+C$$

Linh Vu
Linh Vu
Numerade Educator
02:10

Problem 28

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int \frac{4}{e^{3 x+1}} d x=\frac{k}{e^{3 x+1}}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:42

Problem 29

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int(5 x-7)^{-2} d x=k(5 x-7)^{-1}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:42

Problem 30

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int \sqrt{x+1} d x=k(x+1)^{3 / 2}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:35

Problem 31

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int(4-x)^{-1} d x=k \ln |4-x|+C$$

Linh Vu
Linh Vu
Numerade Educator
01:55

Problem 32

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int \frac{7}{(8-x)^{4}} d x=\frac{k}{(8-x)^{3}}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:48

Problem 33

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int(3 x+2)^{4} d x=k(3 x+2)^{5}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:36

Problem 34

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int(2 x-1)^{3} d x=k(2 x-1)^{4}+C$$

Linh Vu
Linh Vu
Numerade Educator
01:22

Problem 35

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int \frac{3}{2+x} d x=k \ln |2+x|+C$$

Linh Vu
Linh Vu
Numerade Educator
01:35

Problem 36

Find the value of $k$ that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How? ]
$$\int \frac{5}{2-3 x} d x=k \ln |2-3 x|+C$$

Linh Vu
Linh Vu
Numerade Educator
00:22

Problem 37

Find all functions $f(t)$ with the following property:
$$f^{\prime}(t)=t^{3 / 2}$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:15

Problem 38

Find all functions $f(t)$ with the following property:
$$f^{\prime}(t)=\frac{4}{6+t}$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
00:15

Problem 39

Find all functions $f(t)$ with the following property:
$$f^{\prime}(t)=0$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
00:34

Problem 40

Find all functions $f(t)$ with the following property:
$$f^{\prime}(t)=t^{2}-5 t-7$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:52

Problem 41

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=.5 e^{-.2 x}, f(0)=0$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:21

Problem 42

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=2 x-e^{-x}, f(0)=1$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
00:52

Problem 43

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=x, f(0)=3$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:23

Problem 44

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=8 x^{1 / 3}, f(1)=4$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:49

Problem 45

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=\sqrt{x}+1, f(4)=0$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:17

Problem 46

Find all functions $f(x)$ with the following properties:
$$f^{\prime}(x)=x^{2}+\sqrt{x}, f(1)=3$$

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:22

Problem 47

Figure 4 shows the graphs of several functions $f(x)$ for which $f^{\prime}(x)=\frac{2}{x} .$ Find the expression for the function $f(x)$ whose graph passes through $(1,2)$.

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 48

Figure 5 shows the graphs of several functions $f(x)$ for which $f^{\prime}(x)=\frac{1}{3} .$ Find the expression for the function $f(x)$ whose graph passes through $(6,3)$.

Linh Vu
Linh Vu
Numerade Educator
02:03

Problem 49

Which of the following is $\int \ln x d x ?$
(a) $\frac{1}{x}+C$
(b) $x \cdot \ln x-x+C$
(c) $\frac{1}{2} \cdot(\ln x)^{2}+C$

Linh Vu
Linh Vu
Numerade Educator
03:10

Problem 50

Which of the following is $\int x \sqrt{x+1} d x ?$
(a) $\frac{2}{5}(x+1)^{5 / 2}-\frac{2}{3}(x+1)^{3 / 2}+C$
(b) $\frac{1}{2} x^{2} \cdot \frac{2}{3}(x+1)^{3 / 2}+C$

Linh Vu
Linh Vu
Numerade Educator
01:22

Problem 51

Figure 6 contains the graph of a function $F(x) .$ On the same coordinate system, draw the graph of the function $G(x)$ having the properties $G(0)=0$ and $G^{\prime}(x)=F^{\prime}(x)$ for each $x$.

Brad Bailey
Brad Bailey
Numerade Educator
01:09

Problem 52

Figure 7 contains an antiderivative of the function $f(x)$. Draw the graph of another antiderivative of $f(x)$.

Linh Vu
Linh Vu
Numerade Educator
00:43

Problem 53

The function $g(x)$ in Fig. 8 resulted from shifting the graph of $f(x)$ up 3 units. If $f^{\prime}(5)=\frac{1}{4}$, what is $g^{\prime}(5)$ ?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:23

Problem 54

The function $g(x)$ in Fig. 9 resulted from shifting the graph of $f(x)$ up 2 units. What is the derivative of $h(x)=g(x)-f(x) ?$

Linh Vu
Linh Vu
Numerade Educator
04:45

Problem 55

A ball is thrown upward from a height of 256 feet above the ground, with an initial velocity of 96 feet per second. From physics it is known that the velocity at time $t$ is $v(t)=96-32 t$ feet per second.
(a) Find $s(t)$, the function giving the height of the ball at time $t$
(b) How long will the ball take to reach the ground?
(c) How high will the ball go?

Erik Keohane
Erik Keohane
Numerade Educator
03:03

Problem 56

A rock is dropped from the top of a 400 -foot cliff. Its velocity at time $t$ seconds is $v(t)=-32 t$ feet per second.
(a) Find $s(t)$, the height of the rock above the ground at time $t$.
(b) How long will the rock take to reach the ground?
(c) What will be its velocity when it hits the ground?

Erik Keohane
Erik Keohane
Numerade Educator
01:17

Problem 57

Let $P(t)$ be the total output of a factory assembly line after $t$ hours of work. If the rate of production at time $t$ is $P^{\prime}(t)=60+2 t-\frac{1}{4} t^{2}$ units per hour, find the formula for $P(t)$.

Linh Vu
Linh Vu
Numerade Educator
01:42

Problem 58

After $t$ hours of operation, a coal mine is producing coal at the rate of $C^{\prime}(t)=40+2 t-\frac{1}{5} t^{2}$ tons of coal per hour. Find a formula for the total output of the coal mine after $t$ hours of operation.

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:54

Problem 59

A package of frozen strawberries is taken from a freezer at $-5^{\circ} \mathrm{C}$ into a room at $20^{\circ} \mathrm{C}$. At time $t$, the average temperature of the strawberries is increasing at the rate of $T^{\prime}(t)=10 e^{-.4 t}$ degrees Celsius per hour. Find the temperature of the strawberries at time $t$.

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:29

Problem 60

A flu epidemic hits a town. Let $P(t)$ be the number of persons sick with the flu at time $t$, where time is measured in days from the beginning of the epidemic and $P(0)=100$. After $t$ days, if the flu is spreading at the rate of $P^{\prime}(t)=120 t-3 t^{2}$ people per day, find the formula for $P(t)$.

Linh Vu
Linh Vu
Numerade Educator
02:23

Problem 61

A small tie shop finds that at a sales level of $x$ ties per day its marginal profit is $M P(x)$ dollars per tie, where $M P(x)=1.30+.06 x-.0018 x^{2} .$ Also, the shop will lose $$\$ 95$$ per day at a sales level of $x=0 .$ Find the profit from operating the shop at a sales level of $x$ ties per day.

Linh Vu
Linh Vu
Numerade Educator
01:03

Problem 62

A soap manufacturer estimates that its marginal cost of producing soap powder is $C^{\prime}(x)=.2 x+1$ hundred dollars per ton at a production level of $x$ tons per day. Fixed costs are $$\$ 200$$ per day. Find the cost of producing $x$ tons of soap powder per day.

Stark Ledbetter
Stark Ledbetter
Numerade Educator
02:09

Problem 63

The United States has been consuming iron ore at the rate of $R(t)$ million metric tons per year at time $t$, where $t=0$ corresponds to 1980 and $R(t)=94 e^{.016 t} .$ Find a formula for the total U.S. consumption of iron ore from 1980 until time $t$.

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:56

Problem 64

Since 1987, the rate of production of natural gas in the United States has been approximately $R(t)$ quadrillion British thermal units per year at time $t$, with $t=0$ corresponding to 1987 and $R(t)=17.04 e^{.016 t}$. Find a formula for the total U.S. production of natural gas from 1987 until time $t$.

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:35

Problem 65

Drilling of an oil well has a fixed cost of $\$ 10,000$ and a marginal cost of $C^{\prime}(x)=1000+50 x$ dollars per foot. where $x$ is the depth in feet. Find the expression for $C(x)$, the total cost of drilling $x$ feet. $[$ Note: $C(0)=10,000 .$ ]

Stark Ledbetter
Stark Ledbetter
Numerade Educator
01:33

Problem 66

Find an antiderivative of $f(x)$, call it $F(x)$, and compare the graphs of $F(x)$ and $f(x)$ in the given window to check that the expression for $F(x)$ is reasonable. [That is, determine whether the two graphs are consistent. When $F(x)$ has a relative extreme point, $f(x)$ should be zero; when $F(x)$ is increasing, $f(x)$ should be positive, and so on.]
$f(x)=2 x-e^{-.02 x},[-10,10]$ by $[-20,100]$

Christopher Stanley
Christopher Stanley
Numerade Educator
10:29

Problem 67

Find an antiderivative of $f(x)$, call it $F(x)$, and compare the graphs of $F(x)$ and $f(x)$ in the given window to check that the expression for $F(x)$ is reasonable. [That is, determine whether the two graphs are consistent. When $F(x)$ has a relative extreme point, $f(x)$ should be zero; when $F(x)$ is increasing, $f(x)$ should be positive, and so on.]
$f(x)=e^{2 x}+e^{-x}+\frac{1}{2} x^{2},[-2.4,1.7]$ by $[-10,10]$

Kaitlin Yaeger
Kaitlin Yaeger
Numerade Educator
10:29

Problem 68

Find an antiderivative of $f(x)$, call it $F(x)$, and compare the graphs of $F(x)$ and $f(x)$ in the given window to check that the expression for $F(x)$ is reasonable. [That is, determine whether the two graphs are consistent. When $F(x)$ has a relative extreme point, $f(x)$ should be zero; when $F(x)$ is increasing, $f(x)$ should be positive, and so on.]
Plot the graph of the solution of the differential equation $y^{\prime}=e^{-x^{2}}, y(0)=0 .$ Observe that the graph approaches the value $\sqrt{\pi} / 2 \approx .9$ as $x$ increases.

Kaitlin Yaeger
Kaitlin Yaeger
Numerade Educator