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Chapter 5

Integrals

Educators

+ 7 more educators

Problem 1

Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."

Amrita B.
Numerade Educator

Problem 2

Let $ \displaystyle g(x) = \int^x_0 f(t) \,dt $, where $ f $ is the function whose graph is shown.

(a) Evaluate $ g(x) $ for $ x $ = 0, 1, 2, 3, 4, 5, and 6.
(b) Estimate $ g(7) $.
(c) Where does $ g $ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $ g $.

Chris T.
Numerade Educator

Problem 3

Let $g(x)=\int_{0}^{x} f(t) d t,$ where $f$ is the function whose graph is shown.
(a) Evaluate $g(x)$ for $x=0,1,2,3,4,5,$ and 6 .
(b) Estimate $g(7)$
(c) Where does $g$ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $g$.

Chris T.
Numerade Educator

Problem 4

Let $ \displaystyle g(x) = \int^x_0 f(t) \,dt $, where $ f $ is the function whose graph is shown.

(a) Evaluate $ g(0) $ and $ g(6) $
(b) Estimate $ g(x) $ for $ x $ = 1, 2, 3, 4, and 5.
(c) On what interval is $ g $ increasing?
(d) Where does $ g $ have a maximum value?
(e) Sketch a rough graph of $ g $.
(f) Use the graph in part (e) to sketch the graph of $ g'(x) $. Compare with the graph of $ f $.

Chris T.
Numerade Educator

Problem 5

Sketch the area represented by $ g(x) $. Then find $ g'(x) $ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$ \displaystyle g(x) = \int^x_1 t^2\,dt $

Leon D.
Numerade Educator

Problem 6

Sketch the area represented by $ g(x) $. Then find $ g'(x) $ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$ \displaystyle g(x) = \int^x_0 (2 + \sin t)\,dt $

Chris T.
Numerade Educator

Problem 7

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(x) = \int^x_0 \sqrt{t + t^3} \,dt $

Yuki H.
Numerade Educator

Problem 8

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(x) = \int^x_1 \ln (1 + t^2) \,dt $

Yuki H.
Numerade Educator

Problem 9

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(s) = \int^s_5 (t - t^2)^8 \,dt $

Amrita B.
Numerade Educator

Problem 10

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(u) = \int^u_0 \frac{\sqrt{t}}{t + 1} \,dt $

Amrita B.
Numerade Educator

Problem 11

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle F(x) = \int^0_x \sqrt{1 + \sec t} \,dt $

$$ \biggl[ \textit{Hint:} \int^0_x \sqrt{1 + \sec t} \,dt = - \int^x_0 \sqrt{1 + \sec t} \,dt \biggr] $$

Amrita B.
Numerade Educator

Problem 12

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle R(y) = \int^2_y t^3 \sin t \,dt $

Amrita B.
Numerade Educator

Problem 13

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(x) = \int^{e^x}_1 \ln t \,dt $

Amrita B.
Numerade Educator

Problem 14

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(x) = \int^{\sqrt{x}}_1 \frac{z^2}{z^4 + 1} \,dz $

Amrita B.
Numerade Educator

Problem 15

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{3x + 2}_1 \frac{t}{1 + t^3} \,dt $

Amrita B.
Numerade Educator

Problem 16

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{x^4}_0 \cos^2 \theta \,d\theta $

Amrita B.
Numerade Educator

Problem 17

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{\pi/4}_{\sqrt{x}} \theta \tan \theta \,d\theta $

Amrita B.
Numerade Educator

Problem 18

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^1_{\sin x} \sqrt{1 + t^2} \,dt $

Chris T.
Numerade Educator

Problem 19

Evaluate the integral.

$ \displaystyle \int^3_1 (x^2 + 2x - 4) \,dx $

MB
Michelle B.
Numerade Educator

Problem 20

Evaluate the integral.

$ \displaystyle \int^1_{-1} x^{100} \,dx $

Amrita B.
Numerade Educator

Problem 21

Evaluate the integral.

$ \displaystyle \int^2_0 \biggl(\frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \biggr) \,dt $

Amrita B.
Numerade Educator

Problem 22

Evaluate the integral.

$ \displaystyle \int^1_0 (1 - 8v^3 + 16v^7) \,dv $

Yuki H.
Numerade Educator

Problem 23

Evaluate the integral.

$ \displaystyle \int^9_1 \sqrt{x} \,dx $

Amrita B.
Numerade Educator

Problem 24

Evaluate the integral.

$ \displaystyle \int^8_1 x^{-2/3} \,dx $

Amrita B.
Numerade Educator

Problem 25

Evaluate the integral.

$ \displaystyle \int^{\pi}_{\pi/6} \sin \theta \,d\theta $

Amrita B.
Numerade Educator

Problem 26

Evaluate the integral.

$ \displaystyle \int^5_{-5} e \,dx $

Yuki H.
Numerade Educator

Problem 27

Evaluate the integral.

$ \displaystyle \int^1_0 (u + 2) (u - 3) \,du $

Amrita B.
Numerade Educator

Problem 28

Evaluate the integral.
$\int_{0}^{4}(4-t) \sqrt{t} d t$

Amrita B.
Numerade Educator

Problem 29

Evaluate the integral.

$ \displaystyle \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx $

Amrita B.
Numerade Educator

Problem 30

Evaluate the integral.

$ \displaystyle \int^2_{-1} (3u - 2)(u + 1) \,du $

Amrita B.
Numerade Educator

Problem 31

Evaluate the integral.

$ \displaystyle \int^{\pi/2}_{\pi/6} \csc t \cot t \,dt $

Amrita B.
Numerade Educator

Problem 32

Evaluate the integral.

$ \displaystyle \int^{\pi/3}_{\pi/4} \csc^2 \theta \,d\theta $

Amrita B.
Numerade Educator

Problem 33

Evaluate the integral.

$ \displaystyle \int^1_0 (1 + r)^3 \,dr $

Amrita B.
Numerade Educator

Problem 34

Evaluate the integral.

$ \displaystyle \int^3_0 (2\sin x - e^x) \,dx $

Amrita B.
Numerade Educator

Problem 35

Evaluate the integral.

$ \displaystyle \int^2_1 \frac{v^3 + 3v^6}{v^4} \,dv $

Amrita B.
Numerade Educator

Problem 36

Evaluate the integral.

$ \displaystyle \int^{18}_1 \sqrt{\frac{3}{z}} \,dz $

Amrita B.
Numerade Educator

Problem 37

Evaluate the integral.

$ \displaystyle \int^1_0 (x^e + e^x) \,dx $

Amrita B.
Numerade Educator

Problem 38

Evaluate the integral.

$ \displaystyle \int^1_0 \cosh t \,dt $

Amrita B.
Numerade Educator

Problem 39

Evaluate the integral.

$ \displaystyle \int^{\sqrt{3}}_{1/\sqrt{3}} \frac{8}{1 + x^2} \,dx $

Amrita B.
Numerade Educator

Problem 40

Evaluate the integral.

$ \displaystyle \int^3_1 \frac{y^3 - 2y^2 - y}{y^2} \,dy $

Amrita B.
Numerade Educator

Problem 41

Evaluate the integral.

$ \displaystyle \int^4_0 2^s \,ds $

Amrita B.
Numerade Educator

Problem 42

Evaluate the integral.

$ \displaystyle \int^{1/\sqrt{2}}_{1/2} \frac{4}{\sqrt{1 - x^2}} \,dx $

Amrita B.
Numerade Educator

Problem 43

Evaluate the integral.

$ \displaystyle \int^{\pi}_0 f(x) \,dx $ where $ f(x) = \left\{
\begin{array}{ll}
\sin x & \mbox{if $ 0 \le x < \pi/2 $}\\
\cos x & \mbox{if $ \pi/2 \le x \le \pi $}
\end{array} \right.$

Amrita B.
Numerade Educator

Problem 44

Evaluate the integral.

$ \displaystyle \int^2_{-2} f(x) \,dx $ where $ f(x) = \left\{
\begin{array}{ll}
2 & \mbox{if $ -2 \le x \le 0 $}\\
4 - x^2 & \mbox{if $ 0 < x \le 2 $}
\end{array} \right.$

Amrita B.
Numerade Educator

Problem 45

Sketch the region enclosed by the given curves and calculate its area.

$ y = \sqrt{x} $, $ y = 0 $, $ x = 4 $

Amrita B.
Numerade Educator

Problem 46

Sketch the region enclosed by the given curves and calculate its area.

$ y = x^3 $, $ y = 0 $, $ x = 1 $

Amrita B.
Numerade Educator

Problem 47

Sketch the region enclosed by the given curves and calculate its area.

$ y = 4 - x^2 $, $ y = 0 $

Leon D.
Numerade Educator

Problem 48

Sketch the region enclosed by the given curves and calculate its area.

$ y = 2x - x^2 $, $ y = 0 $

Amrita B.
Numerade Educator

Problem 49

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sqrt[3]{x} $, $ 0 \le x \le 27 $

Amrita B.
Numerade Educator

Problem 50

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = x^{-4} $, $ 1 \le x \le 6 $

Amrita B.
Numerade Educator

Problem 51

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sin x $, $ 0 \le x \le \pi $

Jacquelyn T.
Numerade Educator

Problem 52

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sec^2 x $, $ 0 \le x \le \pi/3 $

Amrita B.
Numerade Educator

Problem 53

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$ \displaystyle \int^2_{-1} x^3 \,dx $

Amrita B.
Numerade Educator

Problem 54

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$ \displaystyle \int^{2\pi}_{\pi/6} \cos x \,dx $

Amrita B.
Numerade Educator

Problem 55

What is wrong with the equation?

$ \displaystyle \int^1_{-2} x^{-4} \, dx = \frac{x^{-3}}{-3} \Bigg]^1_{-2} = -\frac{3}{8} $

Amrita B.
Numerade Educator

Problem 56

What is wrong with the equation?

$ \displaystyle \int^2_{-1} \frac{4}{x^3} \, dx = -\frac{2}{x^2} \Bigg]^2_{-1} = -\frac{3}{2} $

Yuki H.
Numerade Educator

Problem 57

What is wrong with the equation?

$ \displaystyle \int^{\pi}_{\pi/3} \sec \theta \tan \theta \, d\theta = \sec \theta \Bigg]^{\pi}_{\pi/3} = -3 $

Bobby B.
University of North Texas

Problem 58

What is wrong with the equation?

$ \displaystyle \int^{\pi}_{0} \sec^2 x \, dx = \tan x \Bigg]^{\pi}_{0} = 0 $

Amrita B.
Numerade Educator

Problem 59

Find the derivative of the function.

$ g(x) = \displaystyle \int^{3x}_{2x} \frac{u^2 - 1}{u^2 + 1} \, du $

$ \displaystyle \Biggl[ Hint: \int^{3x}_{2x} f(u) \, du = \int^0_{2x} f(u) \, du + \int^{3x}_0 f(u) \, du \Biggr] $

Amrita B.
Numerade Educator

Problem 60

Find the derivative of the function.

$ g(x) = \displaystyle \int^{1 + 2x}_{1 - 2x} t \sin t \, dt $

Amrita B.
Numerade Educator

Problem 61

Find the derivative of the function.

$ F(x) = \displaystyle \int^{x^2}_{x} e^{t^2} \, dt $

Amrita B.
Numerade Educator

Problem 62

Find the derivative of the function.

$ F(x) = \displaystyle \int^{2x}_{\sqrt{x}} \arctan t \, dt $

Amrita B.
Numerade Educator

Problem 63

Find the derivative of the function.

$ y = \displaystyle \int^{\sin x}_{\cos x} \ln (1 + 2v) \, dv $

Ma. Theresa A.
Numerade Educator

Problem 64

If $ \displaystyle f(x) = \int^x_0 (1 - t^2) e^{t^2} \,dt $, on what interval is $ f $ increasing?

Amrita B.
Numerade Educator

Problem 65

On what interval is the curve $$ y = \displaystyle \int^x_0 \frac{t^2}{t^2 + t + 2} \, dt $$ concave downward?

Amrita B.
Numerade Educator

Problem 66

Let $ \displaystyle F(x) = \int^x_1 f(t) \, dt $, where $ f $ is the function whose graph is shown. Where is $ F $ concave downward?

Amrita B.
Numerade Educator

Problem 67

Let $ \displaystyle F(x) = \int^x_2 e^{t^2} \, dt $. Find an equation of the tangent line to the curve $ y = F(x) $ at the point with $ x $-coordinate 2.

Amrita B.
Numerade Educator

Problem 68

If $ \displaystyle f(x) = \int^{\sin x}_0 \sqrt{1 + t^2} \, dt $ and $ \displaystyle g(y) = \int^y_3 f(x) \, dx $, find $ g''(\pi/6) $.

Darshan M.
Numerade Educator

Problem 69

If $ f(1) = 12 $, $ f' $ is continuous, and $ \displaystyle \int^4_1 f'(x) \, dx = 17 $, what is the value of $ f(4) $?

Amrita B.
Numerade Educator

Problem 70

The error function $$ \displaystyle \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} \, dt $$ is used in probability, statistics, and engineering.
(a) Show that $ \displaystyle \int^b_a e^{-t^2} \, dt = \frac{1}{2} \sqrt{\pi} [ \text{erf}(b) - \text{erf}(a) ] $.
(b) Show that the function $ y = e^{x^2} \text{erf}(x) $ satisfies the differential equation $ y' = 2xy + 2/\sqrt{\pi} $.

Chris T.
Numerade Educator

Problem 71

The Fresnel function $ S $ was defined in Example 3 and graphed in Figures 7 and 8.

(a) At what values of $ x $ does this function have local maximum values?
(b) On what intervals is the function concave upward?
(c) Use a graph to solve the following equation correct to two decimal places:
$$ \displaystyle \int^x_0 \sin (\pi t^2/2) \, dt = 0.2 $$

Frank L.
Numerade Educator

Problem 72

The sine integral function $$ \displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt $$ is important in electrical engineering. [The integrand $ f(t) = (\sin t)/t $ is not defined when $ t = 0 $, but we know that its limit is 1 when $ t \to 0 $. So we defined $ f(0) = 1 $ and this makes $ f $ a continuous function everywhere.]

(a) Draw the graph of $ \text{Si} $.
(b) At what values of $ x $ does this function have local maximum values?
(c) Find the coordinates of the first inflection point to the right of the origin.
(d) Does this function have horizontal asymptotes?
(e) Solve the following equation correct to one decimal place:
$$ \displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1 $$

Frank L.
Numerade Educator

Problem 73

Let $ \displaystyle g(x) = \int^x_0 f(t) \, dt $, where $ f $ is the function whose graph is shown.

(a) At what values of $ x $ do the local maximum and minimum values of $ g $ occur?
(b) Where does $ g $ attain its absolute maximum value?
(c) On what intervals is $ g $ concave downward?
(d) Sketch the graph of $ g $.

Chris T.
Numerade Educator

Problem 74

Let $ \displaystyle g(x) = \int^x_0 f(t) \, dt $, where $ f $ is the function whose graph is shown.

(a) At what values of $ x $ do the local maximum and minimum values of $ g $ occur?
(b) Where does $ g $ attain its absolute maximum value?
(c) On what intervals is $ g $ concave downward?
(d) Sketch the graph of $ g $.

Chris T.
Numerade Educator

Problem 75

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $ [0, 1] $.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \biggl( \frac{i^4}{n^5} + \frac{i}{n^2} \biggr) $

Amrita B.
Numerade Educator

Problem 76

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $ [0, 1] $.

$ \displaystyle \lim_{n \to \infty} \frac{1}{n} \biggl( \sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + \cdots + \sqrt{\frac{n}{n}} \biggr) $

Bobby B.
University of North Texas

Problem 77

Justify (3) for the case $ h < 0 $.

Amrita B.
Numerade Educator

Problem 78

If $ f $ is continuous and $ g $ and $ h $ are differentiable functions, find a formula for
$$ \displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt $$

Amrita B.
Numerade Educator

Problem 79

(a) Show that $ 1 \le \sqrt{1 + x^3} \le 1 + x^3 $ for $ x \ge 0 $.
(b) Show that $ \displaystyle 1 \le \int^1_0 \sqrt{1 + x^3} \, dx \le 1.25 $

Chris T.
Numerade Educator

Problem 80

(a) Show that $ \cos (x^2) \ge \cos x $ for $ 0 \le x \le 1 $.
(b) Deduce that $ \displaystyle \int^{\pi/6}_0 \cos (x^2) \, dx \ge \frac{1}{2} $.

Chris T.
Numerade Educator

Problem 81

Show that $$ 0 \le \int^{10}_5 \frac{x^2}{x^4 + x^2 + 1} \, dx \le 0.1 $$ by comparing the integrand to a simpler function.

Amrita B.
Numerade Educator

Problem 82

Let $ f(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x < 0 $}\\
x & \mbox{if $ 0 \le x \le 1 $}\\
2 - x & \mbox{if $ 1 < x \le 2 $}\\
0 & \mbox{if $ x > 2 $}
\end{array} \right.$
and $$ g(x) = \int^x_0 f(t) \, dt $$

(a) Find an expression for $ g(x) $ similar to the one for $ f(x) $.
(b) Sketch the graphs of $ f $ and $ g $.
(c) Where is $ f $ differentiable? Where is $ g $ differentiable?

Chris T.
Numerade Educator

Problem 83

Find a function $ f $ and a number $ a $ such that
$ \displaystyle 6 + \int^x_a \frac{f(t)}{t^2} \, dt = 2 \sqrt{x} $ for all $ x > 0 $

Amrita B.
Numerade Educator

Problem 84

The area labeled $ B $ is three times the area labeled $ A $. Express $ b $ in terms of $ a $.

Aparna S.
Numerade Educator

Problem 85

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate $ f = f(t) $, where $ t $ is the time measured in months since its last overhaul. Because a fixed cost $ A $ is incurred each time the machine is overhauled, the company wants to determine the optimal time $ T $ (in months) between overhauls.

(a) Explain why $ \displaystyle \int^t_0 f(s) ds $ represents the loss in value of the machine over the period of time $ t $ since the last overhaul.
(b) Let $ C = C(t) $ be given by $$ C(t) = \frac{1}{t} \Biggl[ A + \int^t_0 f(s) ds \Biggr] $$
What does $ C $ represent and why would the company want to minimize $ C $?
(c) Show that $ C $ has a minimum value at the numbers $ t = T $ where $ C(T) = f(T) $.

Chris T.
Numerade Educator

Problem 86

A high-tech company purchases a new computing system whose initial value is $ V $. The system will depreciate at the rate $ f = f(t) $ and will accumulate maintenance costs at the rate $ g = g(t) $, where $ t $ is the time measured in months. The company wants to determine the optimal time to replace the system.
(a) Let $$ C(t) = \frac{1}{t} \int^t_0 [f(s) + g(s)] \, ds $$
Show that the critical numbers of $ C $ occur at the numbers $ t $ where $ C(t) = f(t) + g(t) $.
(b) Suppose that
$ f(t) = \left\{
\begin{array}{ll}
\frac{V}{15} - \frac{V}{450}t & \mbox{if $ 0 < t \le 30 $}\\
0 & \mbox{if $ t > 30 $}
\end{array} \right.$
and $ g(t) = \frac{Vt^2}{12,900} $ $ t > 0 $
Determine the length of time $ T $ for the total depreciation $ \displaystyle D(t) = \int^t_0 f(s) \, ds $ to equal the initial value $ V $.
(c) Determine the absolute minimum of $ C $ on $ (0, T] $.
(d) Sketch the graphs of $ C $ and $ f + g $ in the same coordinate system, and verify the result in part (a) in this case.

Chris T.
Numerade Educator