Thomas' Calculus...Early Transcendentals – 14th Edition Integrals Sigma Notation and Limits of...

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Table of Contents

Early Transcendentals - 14th Edition .............................................. Error! Bookmark not defined.

Integrals ..................................................................................................................................... 2

Sigma Notation and Limits of Finite Sums ............................................................................. 2

The Definite Integral .............................................................................................................. 4

The Fundamental Theorem of Calculus ................................................................................. 5

Indefinite Integrals and the Substitution Method ................................................................. 7

Definite Integral Substitutions and the Area Between Curves ............................................ 12

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2

Early Transcendentals – 14th Edition

Integrals

Sigma Notation and Limits of Finite Sums

Questions

Riemann Sum

1) Evaluate: 4 4 4 4

5

1 2 3limn

n

n→

+ + + +

2) Evaluate: 1 1 1

lim1 2n n n n n→

+ +

+ + +

3) Evaluate: 2 2 2 2 2 2

lim1 2n

n n n

n n n n→

+ + +

+ + +

4) Evaluate: 2 2 2 2 2 2

1 1 1lim

1 2n n n n n→

+ + +

+ + +

5) Evaluate: 3/2

1 2 2limn

n n n

n→

+ + + +

6) Evaluate:

1 2sin sin sin

limn

n

n n n

n→

+ + +

7) Evaluate: 2 3 11

limn n n nn

n

e e e e

n

−

→

+ + + +

8) Evaluate: 1

lim ln 1n

n

nk

k

n→=

+





3

9) Use the Riemann sum definition of integral to evaluate 0

x

xdx

Hint: ( )1 2 3 0.5 1n n n+ + + + = + .


2

0

x dx

Hint: ( )( )2 2 2 2 11 2 3 1 2 1

6n n n n+ + + + = + +


3

0

x dx

Hint: ( )23 3 3 3 21

1 2 3 14

n n n+ + + + = +


sin xdx

Hint: ( ) ( ) ( )

1sin sin

2 2sin sin 2 sin

sin2

n n

n

+ + + + =

13) Use the Riemann sum definition of integral to evaluate ( )5

2

2

2 3x x dx+

Hint: ( ) ( )( )2 2 2 21 11 2 3 1 , 1 2 3 1 2 1

2 6n n n n n n n+ + + + = + + + + + = + +

Answer Key

1) 1

5 2) ln 2 3)

4

4) ( )ln 1 2+ 5) 1.219

6) ( )1 cos 1− 7) ( )1 cos 1− 8) 2ln2 1− 9) 1

2 10)

1

3

11) 1

4 12) 2 13) 109.5





4

The Definite Integral

Questions

Compute the following integrals:

1) ( )4

2

1

2 4 1x x dx− + 2) ( ) ( )4

20

0 1 ; 1

1

x xf x f x

xx

=

3) Let f be a continuous function.

Prove that:

a. If f is even, then ( ) ( )0

2

a a

a

f x dx f x dx−

= .

b. If f is odd, then ( ) 0

a

a

f x dx−

= .

4) Find the average values if the following functions over the given intervals:

a. ( )1

g xx

= on 1,4 .

b. ( )( )

2

3

1f r

r=

+ on 1,6 .

Answer Key

1) 13

2) 5

112

3) Solution in the recoreding.

4) a. 1

ln 43

b. 3

14





5

The Fundamental Theorem of Calculus

Questions

1) Recall the Fundamental Theorem of Calculus: ( ) ( ) ( ) ( )'

x

a

I x f t dt I x f x= = .

Prove the following generalization: ( ) ( )( )

( ) ( )( ) ( )' '

b x

a

I x f t dt I x f b x b x= = .

2) We previously generalized the FTC to: ( ) ( )( )

( ) ( )( ) ( )' '

b x

a

I x f t dt I x f b x b x= = .

Generalize again to prove: ( ) ( )( )

( )

( ) ( )( ) ( ) ( )( ) ( )' ' '

b x

a x

I x f t dt I x f b x b x f a x a x= = − .

3) Differentiate the following functions:

a. ( )2

2

x

tI x e dt−= b. ( )3

2

1

lnx

tI x dt

t=

c. ( )3

2

ln

x x

I x t tdt

+

= d. ( )2

341

x

x

dtI x

t=

+

4) Evaluate the following limits:

a. 0

20

coslim

sin

x

x

tdt

t

x→

b.

2

300

1lim sin

x

xtdt

x→ c. 2

44

lim4

x

t

x

xe dt

x→ −

5) Investigate the function ( ) ( ) ( )4 10

0

1 1

x

F x t t dt= + − , particularly the following features:

a. Domain of definition b. Extrema and intervals of increase/decrease c. Inflection points and intervals of concavity/convexity

Mean Value Theorem for Integrals

6) Find the point(s) c , as in MTV for Integrals, for ( ) 2

1f x

x= on 1,3 .





6

Answer Key

1) Solution in the recording. 2) Solution in the recording.

3) a. ( )2

' xI x e−= b. 4

9 ln x

x c. ( ) ( ) ( )( )3 3 2' ln 3 1I x x x x x x= + + +

d. ( )2

8 12

2 3'

1 1

x xI x

x x= −

+ +

4) a. 1

2 b.

2

3 c. 164e

5) a. All x . b. No extrema, the function is always increasing.

c. Inflection points: -1, 3

7− , 1 concave: 1x − or

31

7x−

convex: 3

17

x− − or 1x

6) 3c =





7

Indefinite Integrals and the Substitution Method

Questions

The Domain of Logarithmic and Exponent Functions

1) a. ( )10

4 1x dx+ b. ( )10

2 2 1x x dx− + c. ( )

5

4

2dx

x −

2) a. 3 4 10x dx− b. 10

2 4dx

x + c.

( )4

1

xdx

x −

3) a. 1

dx

x x− − b.

1 1

xdx

x + + c.

1

1

xdx

x

−

−

4) a. 1

2 10dx

x − b. 2 2

2 1

xdx

x

+

+ c. 2

4 1

16 1

xdx

x

+

−

5) a. .4 1

2

xdx

x

+

+ . b. 4 1

3 2

xdx

x

+

+ c. 2

1

2 3

xdx

x x

−

+ −

6) a. ( )4 1xe dx+

b. 2 3

4

x x x

x

e e edx

e

+ + c.

3 4

14 x

xe dx

e

+

7) a. ( )2

1xe dx+

b. 3 4

14 x

xe dx

e

+

c.

2 32 4 10

5

x x x

xdx

+ +

8) a. 2

1

1 4dx

x+ b.

2

1

4dx

x− c.

2

21

xdx

x−


9) 2

2

0

2 1

1

xdx

x x

+

+ + 10) 3

2

xxe dx−

11) ( )

44

1

ln xdx

x 12) 2

1

cos 4xdx

13) 4

1

4 1x dx−

+ − 14) 2

0

sin

1 cos

x xdx

x

+ 15)

/2 4

4 40

sin

sin cos

xdx

x x

+





8

16) Compute the following integrals:

a. 4

3 5

4

cos xdx

x x−

+

b. 1

2

1

sin 1

1

x

x−

+

+

Integrals – Derivative Contained


17) a. 2

2

1

xdx

x + b. 2

3 1

xdx

x + c. 2

2

4 1

xdx

x x

+

+ +

18) a. cot x dx b. tan x dx c. sin cos

sin cos

x xdx

x x

−

+

19) a. 2

1

x

x

edx

e

+

+ b. 1

lndx

x x c. 2

2

6 4

4

x xdx

x x

+ +

+

20) a. 2

2xe x dx b. 3 2xe x dx c. 22x

xdx

e

21) a. tan

2cos

xedx

x b. arctan

21

xedx

x+ c. cos2

sin cosx

x xdx

e−

22) a. ( )( )2cos 2 1 4x x dx+ b. ( )cos sin cosx x dx c. ( )cos ln x

dxx

23) a. ( )4 3cos 10 1x x dx+ b. ( )2sin 1x x dx+ c. sin x

dxx

24) a. ln x

dxx b.

2

arctan

1

xdx

x+ c. 2

tan

cos

xdx

x

25) a. 2

2

1

xdx

x + b.

cos

2sin

xdx

x c.

1

lndx

x x

26) a. 2 1 2x xdx+ b. 3 24x x dx+ c. ln x

dxx





9

Integration by Substitution

Compute the following integral:

27) 2 4

xdx

x

−

+ 28) ( )5

22 1x x dx+ 29) ( )

2ln x

dxx

30) 2 4xe xdx+

31) 1

x

x

edx

e + 32) 2 1x x dx+

33) 2 4

xdx

x + 34) 3 2 1x x dx+ 35) 5 2 1x x dx+

36) 3

2 4

xdx

x + 37)

5

3 1

xdx

x + 38) 2 34 1x x dx+

39) 3

3 2 4

xdx

x + 40)

( )4

1

lndx

x x 41)

2 3xe x dx

42) ( )

7

241

xdx

x− 43) 21 xe dx+ 44)

2

11 dx

x+

45) xe dx 46) 3 xe dx 47)

3

1dx

x x+

48) ( )arctan x dx





10

Answer Key

1) a. ( )

114 11

4 11

xC

++ b.

( )21

1

21

xC

−+ c.

( )4

1

2C

x− +

−

2) a. ( ) 334 10 4 10

16x x C− − + b. 10 2 4x C+ + c.

( ) ( )2 3

1 1 1 1

2 31 1C

x x− − +

− −

3) a. ( )

1.5

1.51

1.5 1.5

x xC

− − + +

b. ( )

1.52 1

3

xx C

−− + c. 32

3x x C+ +

4) a. 1

ln 2 102

x C− + b. 1

ln 2 12

x x C+ + + c. 1

ln 4 14

x C− +

5) a. 4 7ln 2x x C− + + b. 4 5 2

ln3 9 3

x x C− + + c. 1

ln 2 32

x C− +

6) a. 4 11

4

xe C+ + b. 3 2

1 1 1

3 2x x xC

e e e− − − + c.

41

323

84

xx

e e C−

− +

7) a. 2 21

2

xe C+ + b. 41

323

84

xx

e e C−

− + c. 0.4 3.2 200

ln 0.4 ln 3.2 ln 200

x x x

C+ + +

8) a. ( )1

arctan 22

x C+ b. arcsin2

xC

+

c.

1 1ln

2 1

xx C

x

+− +

−

9) ln 7 10) 3 24 3e e− −− + 11) 51ln 4

5

12) 1 1

1 sin8 1.0622 8

− − =

13) ( )1.5 1.5216 6 7 11.478

3− + + =

14) 2

4

15) 0.785

4

= 16)

2

17) a. 2ln 1x C+ + b. 31ln 1

3x C+ + c. 2ln 4 1x x C+ + +

18) a. ln sin x C+ b. ln cos x C− + c. ln sin cosx x C− + +

19) a. 2 ln 1xe e C+ + b. ln ln x C+ c. 2ln 4x x x C+ + +

20) a. 2xe C+ b.

31

3

xe C+ c. 221

4

xe C−− +

21) a. tan xe C+ b. arctan xe C+ c. cos21

4

xe C− +

22) a. ( )2sin 2 1x C+ + b. ( )sin sin x C+ c. ( )sin ln x C+





11

23) a. ( )41sin 10 1

40x C+ + b. ( )21

cos 12

x C− + + c. 2cos x C− +

24) a. ( )21

ln2

x C+ b. ( )21

arctan2

x C+ c. 21tan

2x C+

25) a. 22 1x C+ + b. 2 2sin x C+ c. 2 ln x C+

26) a. ( )3

2 22

13

x C+ + b. ( )3

3 22

49

x C+ + c. ( )3

22

ln3

x C+

27) ( )21ln 4

2x C− + + 28) ( )

621

2 124

x C+ +

29) ( )31

ln3

x C+ 30) 2 41

2

xe C+ +

31) ( )ln 1xe C+ + 32) ( )3

2 21

13

x C+ +

33) 2 4x C+ + 34) ( ) ( )5 3

2 22 21 1

1 15 3

x x C+ − + +

35) ( ) ( ) ( )7 5 3

2 2 22 2 21 2 1

1 1 12 5 3

x x x C+ − + + + + 36) ( ) ( )3 1

2 22 21

4 4 43

x x C+ − + +

37) ( ) ( )3 1

3 32 22

1 13

x x C+ − + + 38) ( ) ( )9 5

2 22 22 2

1 19 5

x x C+ − + +

39) ( ) ( )5 2

2 23 33

4 3 410

x x C+ − + + 40) ( )

3

1

3 lnC

x

−+

41) ( )221

12

xx e C− + 42) ( )4

4

1 1ln 1

4 1x C

x

+ − +

−

43) 2

2

2

1 1 11 ln

2 1 1

xx

x

ee C

e

+ −+ + +

+ + 44)

22

2

1 1 11 ln

2 1 1

xx C

x

+ −+ + +

+ +

45) ( )2 1 xx e C− + 46) 3

2 1

3 33 2 2xe x x C

− + +

47) 6 66 ln 13 2

x xx x C

− + − + +

48) ( ) ( )1 arctanx x x C+ − +





12

Definite Integral Substitutions and the Area Between Curves

Questions

1) Given two functions:

( ) 2 4 6f x x x= + + , ( ) 2 4 14g x x x= − + .

a. Find their point of intersection. b. Find the area bounded by the graphs of the two

functions, the x -axis, and the lines 2x = and 2x = − (the shaded area in the figure).

2) Given the function 2 6 5y x x= − + − (see figure).

a. Find the coordinates of the maximum point of the function.

Find the equation of tangent to the graph at its maximum point.

Find the area bounded by this tangent, by the axes and by the graph of the function (the shaded area in the figure).

3) Given the function ( ) ( )2

2f x x= − and the line

0.5 0.5y x= + (see figure).

Find the area bounded by the graph of the function, the given line and the x -axis (the shaded area in the figure).

4) Given the functions: ( ) 2f x x= , ( ) 2 18g x x= − + .

Their graphs intersect at points A and B (see figure). Find the x -coordinates of the points A and B. Compute the area, in the first quadrant,

bounded by the functions, by the x -axis and by the line 4x = (The shaded area in the figure).





13

5) Given two functions: 2 3 2y x x= − + + , 3 3 2y x x= − + .

Find the x -coordinates of the intersection points of the graphs of the two functions.

Compute the area bounded by the graphs of the two functions (The shaded area in the figure).

6) Given the function ( ) 2f x x ax= − + . The function

passes through the point ( )A 2,8 (see figure).

Find the parameter a . The function cuts the x -axis at the origin and at

another point B. Find the coordinates of B. Compute the area (shaded in the figure) bounded

by the graph of the function, by the chord AB and by the x -axis.

7) Given the functions: ( ) 2xf x e− += , ( ) xg x e= .

Find the intersection points of the functions with the y -axis.

Find the intersection point of the two functions.

The areas 1S and 2S are as in the figure.

Compute the ratio 1

2

S

S.

8) Given the function ( ) 2xf x e−= . A line to the function

was drawn at the point 1x = − (see figure). Find the equation of the tangent. Compute the area (shaded in the figure) bounded

by the graph of the function, by the tangent and by the axes.





14

9) Given the function cos2y x= on the domain 04

x

(see figure). A tangent to the graph is drawn at the

point, where 4

x

= .

Find the equation of the tangent. Compute the area (shaded in the figure) bounded

by the graph of the function, by the tangent and by the y -axis.

10) Compute the area bounded by the graph of the

function1

2 1y

x=

−, and by the lines 3x = and 1y = .

(The shaded area in the figure).

11) Given the function ( ) 2x xf x e e= − , which has a

minimum as shown in the figure. Find the x -coordinate of this minimum point. A perpendicular is dropped from the minimum

point to the x -axis. The shaded area in the figure is bounded by this perpendicular, the graph of the function, the x -axis and the line x a= .

Given that it has an area of 23 a ae e− , whereln 0.5a , find the value of a .

12) Given the function ( )1

2

x

f x e+

= as in the figure.

The slope of the tangent to the graph at a point A on it is2

2

e.

Find the coordinates of point A. Find the equation of the tangent. Compute the area bounded by: the graph of the function,

the tangent and the y -axis.





15

13) Given the function ( )8

2f xx

= − on the domain 0x .

A tangent line to the graph is drawn at the point

( )A 2,2 , (see figure).

Find the equation of the tangent. Compute the area bounded by: the graph of the

function, the tangent and the x -axis.

14) Answer the following:

Draw a rough sketch of the graphs of the following 2 functions:

( )

( )

sin ; 0

cos 2 ; 0

f x x x

g x x x

=

=

Shade or highlight the area bounded by these and compute its size.

15) Given the function ( ) 2tanf x x= on the domain 02

x

− :

Find the equation of its tangent at the point where4

x

= − .

Show that 2tan tanxdx x x c= − + , and find the area bounded by the graph of the

function, the tangent and the x -axis.

16) Through the point ( )A 8,0 tangents are drawn to the parabola 2 10 25y x x= − + .

Find the equations of the tangents. Compute the area bounded by the tangents and the parabola.

17) Given the function ( ) 4f x x x= + on the domain

0x (see figure). Find the equation of the line passing through

the origin and tangent to the graph of the given function.

Compute the area bounded by the graph of the function, the tangent and the y -axis.





16

18) Answer the following questions:

Compute the derivative of the function ( ) 3cosf x x= .

Compute the area bounded the x -axis and by the graph of the function

2cos siny x x= on the domain1 3

2 2x .

19) Compute the area bounded by the parabola 2y x= − and the line 6y x= + .

20) Compute the area bounded by the parabola 2 2x y= + and the line 8y x= − .

21) Compute the following integrals:

2 2

0

a

x a dx−

2 2

a

a

a y dy−

−





17

Answer Key

1) a. ( )1,11 b. 25.33

2) a. ( )3,4 b. 4y = c. 2

63

3) 1

13

4) a. 3x = b. 2

143

5) a. 0, 3,2x = − b. 3

154

6) a. 6a = b. ( )B 6,0 c. 1

25 3

7) a. ( ) ( )20,1 , 0, e b. ( )1,e c. 1

2

1 S

eS

= −

8) a. 2 22y e x e= − − b. 21 1

4 2e −

9) a. 22

y x

= − + b. 2 1

16 2

−

10) 1

2 ln52

−

11) a. ln 0.5 x = b. ( )1

ln 0.152

a =

12) a. ( )2A 3,e b. 2 2

2 2

e ey x= − c. 2 0.55

24

e e−

13) a. 2 6 y x= − + b. 8ln 2-5

14) b. 3 3

2 a. See the following sketch:

15) a. 4 1y x = − − + b. 7

8 4

−

16) a. 0y = b. 18

17) a. 3y x= b. 4.8

18) a. ( ) 2' 3cos sinf x x x= − b. 2

3

19) 5

620 .

20) 5

620 .

21) a. 20.25 a b. 20.5 a



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