calculus 2 - grant's tutoring · grant’s tutoring (text or call (204) 489-2884) do not...

© Grant Skene for Grant’s Tutoring (www.grantstutoring.com) DO NOT RECOPY

Grant’s Tutoring is a private tutoring organization and is in no way affiliated with the University of Manitoba.

CALCULUS 2 (INTEGRATION and APPLICATIONS)

THE ENTIRE COURSE IN ONE BOOK

© Grant Skene for Grant’s Tutoring (text or call (204) 489-2884) DO NOT RECOPY

Grant’s Tutoring is a private tutoring organization and is in no way affiliated with the University of Manitoba.

HOW TO USE THIS BOOK

I have broken the course up into lessons. Note that all the Lecture Problems for all of the

lessons are at the start of this book (pages 6 to 19). Then I teach each lesson, incorporating

these Lecture Problems when appropriate. Make sure you read my suggestion about

Index Cards on page 5. If you are able to solve all the Practise Problems I have given you,

then you should have nothing to fear about your Midterm or Final Exam.

I have presented the course in what I consider to be the most logical order. Although my

books are designed to follow the course syllabus, it is possible your prof will teach the course in

a different order or omit a topic. It is also possible he/she will introduce a topic I do not cover.

Make sure you are attending your class regularly! Stay current with the

material, and be aware of what topics are on your exam. Never forget, it is your

prof that decides what will be on the exam, so pay attention.

If you have any questions or difficulties while studying this book, or if you believe you

have found a mistake, do not hesitate to contact me. My phone number and website are noted

at the bottom of every page in this book. “Grant’s Tutoring” is also in the phone book.

I welcome your input and questions.

Wishing you much success,

Grant Skene

Owner of Grant’s Tutoring

Have you signed up for

Grant’s Homework Help yet?

No? Then what are you waiting for? IT’S FREE!

Go to www.grantstutoring.com right now,

and click the link to sign up for

Grant’s Homework Help

IT’S FREE!

Grant will send you extra study tips and questions of interest

throughout the term.

You are also welcome to contact Grant with any questions you

have. Your question may even provide the inspiration for other

tips to send.

If there are any changes in the course work or corrections to this

book, you will be the first to know.

You will also be alerted to upcoming exam prep seminars and

other learning aids Grant offers.

If you sign up, you will also receive a coupon towards Grant’s

services.

And, it is all FREE!

http://www.grantstutoring.com/

http://upload.wikimedia.org/wikipedia/commons/a/a7/Stop_sign_MUTCD.svg

Four ways Grant can help you:

Grant’s Study Books

Basic Statistics 1 (Stat 1000)


Linear Algebra and Vector Geometry (Math 1300)

Matrices for Management (Math 1310)

Intro Calculus (Math 1500 or Math 1510)

Calculus for Management (Math 1520)

Calculus 2 (Math 1700 or 1710)

All these books are available at UMSU Digital Copy Centre, room 118

University Centre, University of Manitoba. Grant’s books can be

purchased there all year round. You can also order a book from

Grant directly. Please allow one business day because the books are

made-to-order.

Grant’s One-Day Exam Prep Seminars

These are one-day, 12-hour marathons designed to explain and review all

the key concepts in preparation for an upcoming midterm or final exam.

Don’t delay! Go to www.grantstutoring.com right now to see the date of the

next seminar. A seminar is generally held one or two weeks before the

exam, but don’t risk missing it just because you didn’t check the date well in

advance. You can also reserve your place at the seminar online. You are

not obligated to attend if you reserve a place. You only pay for the seminar

if and when you arrive.

Grant’s Weekly Tutoring Groups

This is for the student who wants extra motivation and help keeping on top

of things throughout the course. Again, go to www.grantstutoring.com for

more details on when the groups are and how they work.

Grant’s Audio Lectures

For less than the cost of 2 hours of one-on-one tutoring, you can listen to

over 40 hours of Grant teaching this book. Hear Grant work through

examples, and offer that extra bit of explanation beyond the written word.

Go to www.grantstutoring.com for more details.




© Grant Skene for Grant’s Tutoring (Questions? Call 489-2884 or go to www.grantstutoring.com) DO NOT RECOPY

TABLE OF CONTENTS

Formulas to Memorize .......................................................................................................1

Index Cards .......................................................................................................................5

Lecture Problems ..............................................................................................................6

Lecture ............................................................................................................................ 20

Lesson 1: Inverse Trigonometric Functions .......................................................... 20

Lesson 2: The Fundamental Theorem of Calculus ................................................. 30

Lesson 3: Riemann Sums ...................................................................................... 43

Lesson 4: The Method of u Substitution................................................................ 56

Lesson 5: Area between 2 Curves ......................................................................... 63

Lesson 6: Volumes ............................................................................................... 74

Lesson 7: Integrals of Trigonometric Functions ................................................... 91

Lesson 8: Integration by Parts ............................................................................. 99

Lesson 9: Integrating by Trig Substitution ......................................................... 107

Lesson 10: Integrating Rational Functions ......................................................... 115

Lesson 11: L’Hôpital’s Rule ................................................................................ 126

Lesson 12: Improper Integrals and the Comparison Theorem ............................ 133

Lesson 13: Arc Length and Surface Area ............................................................ 147

Lesson 14: Parametric Equations ....................................................................... 157

Lesson 15: Polar Curves ..................................................................................... 176

Midterm Exams for Calculus 2 (136.170) ...................................................................... 189

Final Exams for Calculus 2 (136.170) ............................................................................ 205

Solutions to Midterm Exams for Calculus 2 (136.170) ................................................... 229

Solutions to Final Exams for Calculus 2 (136.170) ........................................................ 253

CALCULUS 2 (INTEGRATION) 1


FORMULAS TO MEMORIZE

Elementary Integrals

Standard Integrals

1. K du K du Ku= =∫ ∫

2. 111

n nu du un

+=+∫

3. u ue du e=∫

Note: 1ax axe dx ea

=∫

e.g. 5 515

x xe dx e C= +∫

4. ln

uu a

a dua

=∫

The 3 Fraction Integrals

1. 1

lndu

du uu u

⌠ ⌠

⌡⌡

= =

Note: 1

lndx

ax bax b a

⌠⌡

= ++

e.g. 1

ln 2 52 5 2

dxx C

x⌠⌡

= − +−

2. 12 2

1tan

du ua u a a

−⌠⌡

=+

3. 1

2 2sin

du uaa u

−⌠⌡

=−

Trigonometric Integrals

1. sin cosudu u= −∫

Note: 1

sin cosax dx axa

= −∫

e.g. ( ) ( )1sin 3 cos 3

3x dx x C= − +∫

2. cos sinudu u=∫

Note: 1

cos sinax dx axa

=∫

e.g. ( ) ( )1cos 5 sin 5

5x dx x C= +∫

3. tan ln secudu u=∫

4. sec ln sec tanudu u u= +∫

5. cot ln sinudu u=∫

6. csc ln csc cotudu u u= −∫

7. 2sec tanudu u=∫

8. 2csc cotudu u= −∫

9. sec tan secu udu u=∫

10. csc cot cscu udu u= −∫

2 CALCULUS 2 (INTEGRATION)


Differentiation Rules

The Power Rule: ( ) 1n nx nx −′ = The Product Rule: ( )f g f g f g′ ′ ′⋅ = +

The Quotient Rule: 2

T T B T BB B

′ ′ ′− =

Chain Rule Version of Power Rule: ( ) 1n nu nu u−′ ′= ⋅

Derivatives of Trigonometric Functions:

( )sin cosu u u′ ′= ⋅ ( ) 2tan secu u u′ ′= ⋅ ( )sec sec tanu u u u′ ′= ⋅

( )cos sinu u u′ ′= − ⋅ ( ) 2cot cscu u u′ ′= − ⋅ ( )csc csc cotu u u u′ ′= − ⋅

Derivatives of Exponential and Logarithmic Functions:

( )u ue e u′ ′= ⋅ ( ) lnu ua a u a′ ′= ⋅ ⋅ ( )lnu

uu′′ = ( ) ( )

loglna

uu

u a′′ =

Derivatives of Inverse Trigonometric Functions:

( )1

2

1sin

1u u

u− ′

′= ⋅−

( )1

2

1cos

1u u

u− ′ − ′= ⋅

− ( )1

2

1tan

1u u

u− ′

′= ⋅+

The Definition of the Definite Integral:

( ) ( ) ( )0

1 1

lim * limb n n

iP ni ia

b a ib af x dx f x x f a

n n→ →∞= =

− −= ∆ = ⋅ +

∑ ∑∫

Summation Formulas to Memorize:

1

1n

i

n=

=∑ ( )1

12

n

i

n ni

=

+=∑

( )( )2

1

1 2 16

n

i

n n ni

=

+ +=∑

Fundamental Theorem of Calculus: ( ) ( )u

af t dt f u u

′ ′= ⋅∫



Trigonometric Values to Memorize

θ 0 6π

4π

3π

2π

sinθ 0 12

22

32

1

cosθ 1 32

22

12

0

tan θ 0 13

1 3 undefined

Trigonometric Identities to Memorize

Pythagorean Identities:

2 2sin cos 1θ θ+ = 2 2tan 1 secθ θ+ = 2 2cot 1 cscθ θ+ =

Half-Angle Identities:

( )2 1cos 1 cos2

2θ θ= +

( )2 1sin 1 cos2

2θ θ= −

sin2 2sin cosθ θ θ=

Area between Two Curves: ( )1 2

x b

x a

y y dx=

=

−∫ or ( )1 2

y b

y a

x x dy=

=

−∫

Volume of a Solid of Revolution:

If you wish to use “dx” and are told to revolve about the x-axis;

or, if you wish to use “dy” and are told to revolve about the y-axis; then you will use the “disk” or “washer” method:

Washer Method: ( )2 21 2

x b

x a

V y y dxπ=

=

= −∫ or ( )2 21 2

y b

y a

V x x dyπ=

=

= −∫

If you wish to use “dx” and are told to revolve about the y-axis;

or, if you wish to use “dy” and are told to revolve about the x-axis; then you will use the “cylindrical shell” method:

Cylindrical Shell Method: ( )1 22x b

x a

V x y y dxπ=

=

= −∫ or ( )1 22y b

y a

V y x x dyπ=

=

= −∫



Average Value of f(x) from a to b: ( )1 b

a

f x dxb a− ∫

Arc Length:

Let L = the length of a curve from a to b, then b

a

L dL= ∫ , where there are several formulas

for dL:

Cartesian (x, y) curves: 2

1dy

dL dxdx

= +

or 2

1dx

dL dydy

= +

Parametric curves: 2 2dx dy

dL dtdt dt

= +

Polar curves: 2

2 drdL r d

dθ

θ = +

Surface Area:

Let S = the surface area of a curve from a to b rotated about the given axis, then:

If the curve is rotated about the x-axis: 2b

a

S y dLπ= ∫

If the curve is rotated about the y-axis: 2b

a

S x dLπ= ∫

Area of a Polar Curve:

Let A = the area of a polar curve from θ = a to θ = b, then: 212

b

a

A r dθ= ∫



INDEX CARDS

To make sure that you learn to recognize what technique of integration is

required at the sight of an integral, I suggest that you make 3 by 5 index cards for

Lessons 2 (question 1 only), 4, 7, 8, 9 and 10. As you are doing the homework

for these lessons put each question on a separate index card. On the front of the

card put the integral question, on the back put a note as to where the question

came from. Put just one question per card and, on the front, do not in anyway

indicate where the question came from (put that information on the back). Do

this for the questions in the lesson and the questions I have suggested from the

old exams.

As you amass more and more index cards you will be able to shuffle them

up. This way, as you look at an integral on the index card, you won’t know what

lesson it came from. Thus, you will have to be able to identify the technique

required to solve it just by looking at the integral itself.

BE SURE TO MAKE THESE CARDS. BELIEVE ME IT WORKS!

Most students think they know the techniques, but don’t realize that the

only reason they knew to use Integration by Parts, for example, to solve a given

question was because they were doing the homework in the Integration by Parts

lesson. Students who have made index cards have become much more proficient

at solving integrals without any hint as to what technique is required.



Lecture Problems

Lesson 1: Inverse Trigonometric Functions

Trigonometric Values to Memorize

θ 0 6π

4π

3π

2π

sinθ 0 12

22

32

1

cosθ 1 32

22

12

0

tan θ 0 13

1 3 undefined

Derivatives of Inverse Trigonometric Functions:

( )1

2

1sin

1u u

u− ′

′= ⋅−

( )1

2

1cos

1u u

u− ′ − ′= ⋅

− ( )1

2

1tan

1u u

u− ′

′= ⋅+

1. Evaluate the following:

(a) 1 1sin

2−

(b) 1 1sin

2− −

(c) 1 3cos

2−

(d) 1 2cos

2−

−

(e) ( )1tan 3− − (f) 1 1tan cos

2−

(g) 1 1sin cos

3− −

(h) ( )1cos sin 0.7−

(i) 1sin sin5π−

(j) 1 3tan tan

4π−

2. Compute dydx

for the following (do not simplify):

(a) 1siny x x−= (b) ( )( )

1 5sin

tan 3

xy

x

−

=

(c) 1

2

3tan

5y

x−=

− (d) ( )1 3cosy x x−= +



Homework for Lesson 1: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 189 #1; p. 190 #2; p.191, #3;

p. 192 #2; p. 193 #3; p. 194 #2 (b); p. 195 #9; p. 196 #5; p. 197 #5; p. 198 #2; p. 199 #2; p. 202 #1 (a); p. 203 #4 (a); p. 211 #1; p. 213 #2; p. 224 #5 (a) and (b). Note: the solutions to these questions begin on page 229.

Lesson 2: The Fundamental Theorem of Calculus

( ) ( )u

af t dt f u u⌠

⌡

′= ⋅ ′

Average Value of f(x) from a to b: ( )1 b

a

f x dxb a− ∫

Be sure to memorize all the Elementary Integrals on page 1.

1. Solve the following definite and indefinite integrals:

(a) ( )5

2cos 3x dx

x

⌠⌡

+

(b) 3 22

21 32

xe x dxx x

⌠⌡

+ + −

(c)

2

2

1

5 32

sinx dx

x x

⌠⌡

+ −

(d) ( )( )24

0

1x x dx+∫

(e) ( )4

0sin sec4 tan4x x x dx

π

+∫ (f) 25 csc

3

x xdx

⌠⌡

− +

e

(g) ( )4

0

f x dx∫ where ( )2

3

3 if 1

2 if 1

x xf x

x x

<=

+ ≥ (h)

32

1

4x dx−

−∫

2. Find ( )f x′ for the following functions:

(a) ( ) 3

11

xdt

f xt

⌠⌡

=+

(b) ( ) 35

2cos

xf x t dt⌠

⌡=

(c) ( )2

2

2

sinx

tf x x e dt= ∫ (d) ( )( )

3

7

3

ln

59

x

x

yf x dy

y

⌠⌡

+=

+



Homework for Lesson 2: While doing this homework make index cards for question 1 only (see page 5 for an

explanation). Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 189 #2; p.191, #1; p. 193 #2;

p. 194 #2 (a); p. 195 #6; p. 196 #2 and #3; p. 198 #1 (e) and #3; p. 199 #1 (e) and #3; p. 200 #2 and #5; p. 201 #3; p. 202 #1 (b); p. 203 #4 (b) and (c); p. 205 #1 (a); p. 209 #2; p. 214 #9 Part B; p. 216 #9; p. 217 #3. Note: the solutions to these questions begin on page 229.

Lesson 3: Riemann Sums The Definition of the Definite Integral:

( ) ( ) ( )0

1 1

lim * limb n n

iP ni ia

b a ib af x dx f x x f a

n n→ →∞= =

− −= ∆ = ⋅ +

∑ ∑∫

Summation Formulas to Memorize:

1

1n

i

n=

=∑ ( )1

12

n

i

n ni

=

+=∑

( )( )2

1

1 2 16

n

i

n n ni

=

+ +=∑

1. Compute the left and right Riemann sums for the following functions over the given

interval using 4 equal partitions. Include a sketch of the region. (a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]

2. Compute the lower and upper Riemann sums for the following functions over the given

interval using 4 equal partitions. Include a sketch of the region. (a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]

3. Compute the Riemann sum for the following functions over the given interval using n

equal partitions. Then determine the limit as n approaches infinity (n →∞ ). Check your answer by computing the associated definite integral.

(a) ( ) 23 4f x x= − on [0, 4] (b) ( ) 2 4 3f x x x= − + on [–1, 7]



4. Solve the following limits:

(a) 1

1lim

n

ni

in n→∞

=∑ (b)

5

1

3 3lim 2

n

ni

in n→∞

=

+

∑

(c) 1

lim sinn

ni

in nπ π

→∞=∑ (d) 2

1

2 1lim

21 1

n

ni n i

n

→∞=

⋅ + +

∑

Homework for Lesson 3: Repeat questions 1 to 4 in this lesson. Do the following questions from the old exams: p. 194 #5; p. 197 #1; p. 200 #1;

p. 201 #1; p. 202 #4. Note: the solutions to these questions begin on page 229. Lesson 4: The Method of u Substitution 1. Evaluate the following:

(a) ( )324

5

1

xdx

x

⌠⌡ −

(b) 43 24 xx e dx∫

(c) 2ln lnx x

dxx

⌠⌡

+ (d)

2

62x dx

x

⌠⌡ +

(e) 35 3 1x x dx+∫ (f) 2 6 10dx

x x⌠⌡ + +

(g) 24 2

dx

x x

⌠⌡ + −

(h) ( )

2

3 x

x

e dx

e

⌠⌡

−

(i) 2

1 3

1 8

1 11 dx

x x

⌠⌡

+ (j) 2 4 13xdx

x x+ +⌠⌡



Homework for Lesson 4: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #3 and #6; p. 190 #1 (a)

and (d); p.191, #2 (a); p. 192 #1 (a) and (b) and #3; p. 193 #1 (a) and (b); p. 194 #1 (a); p. 195 #1; p. 196 #1 (b); p. 197 #2; p. 198 #1 (a) and (f); p. 199 #1 (a) and (b); p. 200 #3 (a); p. 201 #2 (a) and (b); p. 202 #2 (a); p. 203 #1 (a) and #2 (a); p. 205 #1 (b) and (c); p. 207 #1 (b); p. 209 #3 (a); p. 211 #2 (b); p. 213 #3 (a) and (b); p. 214 #9 Part A; p. 215 #3 (a), (b) and (d); p. 221 #1 (a); p. 223 #1 (a) and (c); p. 225 #1 (a); p. 227 #1 (a) and (c). Note: the solutions to these questions begin on page 229.

Lesson 5: Area between 2 Curves

( )1 2

x b

x a

y y dx=

=

−∫ or ( )1 2

y b

y a

x x dy=

=

−∫

1. Compute the areas of the following bounded regions: (a) 3y x= and 1 3y x= (b) 2x y= and 22 2x y y= − −

(c) 1

yx

= and 2 2 5x y+ = (d) 212

y x= and 2

11

yx

=+

(e) 5xy e= , x = 0, x = 2, and the x-axis (f) 2 consecutive intersections of siny x= and cosy x= 2. Let R be the region inside the circle centred at the origin of radius 3 and above the line

2 3y x= − . Set up BUT DO NOT EVALUATE the integral (or integrals) that would find the area of R.

Homework for Lesson 5: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 190 #4 (a) and (b) part (i);

p.191, #4; p. 192 #4 (a) and (b); p. 193 #4 (a) and (b); p. 195 #5; p. 196 #4; p. 198 #4; p. 199 #4; p. 200 #4; p. 201 #4; p. 202 #5; p. 203 #3; p. 209 #1; p. 211 #5 (a); p. 213 #1 (a) and (b); p. 215 #1 (a) and (b); p. 217 #2; p. 221 #4 (a); p. 228 #6 (a). Note: the solutions to these questions begin on page 229.



Lesson 6: Volumes

Washer Method: ( )2 21 2

x b

x a

V y y dxπ=

=

= −∫ or ( )2 21 2

y b

y a

V x x dyπ=

=

= −∫

Cylindrical Shell Method: ( )1 22x b

x a

V x y y dxπ=

=

= −∫ or ( )1 22y b

y a

V y x x dyπ=

=

= −∫

1. Compute the volumes of the following bounded regions, if the region is rotated about: (i) the x-axis (ii) the y-axis (a) 3y x= and 2y x= (b) y x= and 24x y y= − (c) sin 2y x= + and 2y = , 0 x π≤ ≤ (d) 2xy e= , the x-axis, x = 1 and x = 2 2. Given the region bounded by y x= and 2y x= , find the volume if this region is rotated

about the x-axis by using: (a) the “disk” or “washer” method (b) the cylindrical shell method. 3. For the region R bounded by 24y x x= − and 28 2y x x= − , set up integrals for the

following: (a) The volume of the solid created by rotating R about the x-axis. (b) The volume of the solid created by rotating R about the y-axis. (c) The volume of the solid created by rotating R about the line x = –2. (d) The volume of the solid created by rotating R about the line y = 10. (e) The volume of the solid created by rotating R about the line x = 7. 4. Derive the formula for the volume of a right circular cone of height h and base radius r

by setting up and solving a volume integral. 5. Derive the formula for the volume of a sphere of radius r by solving the appropriate

volume integral. 6. A solid has a circular base of radius 3 units. Find the volume of the solid if parallel cross-

sections perpendicular to the base are squares.



Homework for Lesson 6: Repeat questions 1 to 6 in this lesson. Do the following questions from the old exams: p. 190 #3 and #4 (skip part (i));

p.191, #5; p. 192 #4 (c) and #5; p. 193 #4 (skip (b)); p. 194 #4 and #5; p. 195 #7 and #8; p. 196 #6; p. 197 #3; p. 206 #5 (c); p. 207 #4 (c) and (d); p. 210 #6 (c); p. 211 #5 (b); p. 213 #1 (c); p. 215 #1 (c); p. 217 #1; p. 219 #3 and #4; p. 221 #4 (c); p. 224 #6 and #7; p. 226 #5 (a) and (b) and #6; p. 227 #5; p. 228 #6 (b) and (c). Note: the solutions to these questions begin on page 229.

Lesson 7: Integrals of Trigonometric Functions

Trigonometric Identities to Memorize

Pythagorean Identities: 2 2sin cos 1θ θ+ = 2 2tan 1 secθ θ+ = 2 2cot 1 cscθ θ+ =

Half-Angle Identities:

( )2 1cos 1 cos2

2θ θ= +

( )2 1sin 1 cos2

2θ θ= −

sin2 2sin cosθ θ θ= 1. Evaluate the following: (a) 2cos 2x dx∫ (b) 2 2sin cosx x dx∫

(c) 3sin 2x dx∫ (d) 3 6sin cosx x dx∫

(e) 2 32sin cosx x dx∫ (f) 4tan secx x dx∫

(g) 2

2

1 cotcos

xdx

x

⌠⌡

− (h) ( )21 cot x dx+∫

Homework for Lesson 7: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #4; p. 190 #1 (b); p.191,

#2 (c); p. 192 #1 (d); p. 193 #1 (d); p. 194 #1 (b); p. 195 #2; p. 196 #1 (a); p. 197 #4 (b); p. 198 #1 (b); p. 200 #3 (b); p. 201 #2 (e); p. 202 #3 (a); p. 203 #2 (b); p. 207 #1 (c); p. 211 #2 (a); p. 217 #4 (b); p. 219 #1 (d); p. 225 #1 (c). Note: the solutions to these questions begin on page 229.



Lesson 8: Integration by Parts

udv uv vdu= −∫ ∫

1. Evaluate the following: (a) 2 lnx x dx∫ (b) ( )2

ln x dx∫

(c) 1sin 2x dx−∫ (d) 21

0

xx e dx∫

(e) ( )2 sin 4x x dx∫ (f) 2secx x dx∫

(g) 2 cos3xe x dx∫ (h) ( )cos ln x dx∫

Homework for Lesson 8: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #7; p. 190 #1 (c); p.191,

#2 (b); p. 192 #1 (c); p. 193 #1 (c); p. 194 #1 (d); p. 195 #3; p. 196 #1 (d); p. 197 #4 (a); p. 198 #1 (d); p. 199 #1 (d); p. 200 #3 (c); p. 201 #2 (c); p. 202 #3 (b); p. 203 #1 (b); p. 205 #2 (a); p. 207 #1 (d); p. 209 #3 (c); p. 213 #3 (c) and (d); p. 215 #3 (c); p. 217 #4 (a); p. 219 #1 (b); p. 221 #1 (e); p. 223 #1 (b); p. 225 #1 (b); p. 227 #1 (b). Note: the solutions to these questions begin on page 229.

Lesson 9: Integrating by Trig Substitution 1. Evaluate the following:

(a) 2

2

1 xdx

x

⌠⌡

− (b)

2 24

dx

x x

⌠⌡ −

(c) ( )3 22 9

dx

x

⌠⌡ +

(d) 24 9

dx

x x

⌠⌡ −

(e) ( )221 9

dx

x

⌠⌡ +

(f) ( )22 2 2

dx

x x

⌠⌡ + +



Homework for Lesson 9: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #5; p.191, #2 (d); p. 193

#1 (e); p. 194 #1 (c); p. 196 #1 (c); p. 198 #1 (c); p. 199 #1 (c); p. 200 #3 (d); p. 201 #2 (d); p. 202 #2 (b); p. 203 #1 (c); p. 205 #2 (c); p. 211 #2 (c); p. 213 #3 (e); p. 215 #3 (e); p. 217 #5; p. 219 #1 (c); p. 221 #1 (b); p. 223 #1 (d); p. 227 #1 (d). Note: the solutions to these questions begin on page 229.

Lesson 10: Integrating Rational Functions 1. Evaluate the following:

(a) 11

xdx

x⌠⌡

−+

(b) 2 4 5

2x x

dxx

⌠⌡

+ +−

(c) 2

26 7

xdx

x x

⌠⌡

+− −

(d) ( )( )( )

2 3 41 2 3x x

dxx x x

⌠⌡

+ +− − −

(e) 4 1x

dxx

⌠⌡ −

(f) ( )( )21 4

dxx x

⌠⌡ − +

(g) ( ) ( )2 2

2 3

dx

x x

⌠⌡ − −

(h) 3 2

dxx x

⌠⌡ −

Homework for Lesson 10: While doing this homework make index cards (see page 5 for an explanation). Repeat question 1 in this lesson. Do the following questions from the old exams: p. 189 #8; p. 202 #2 (c); p. 205

#2 (b); p. 207 #1 (a); p. 209 #3 (b); p. 211 #2 (d); p. 213 #3 (f); p. 215 #3 (f); p. 217 #4 (c); p. 219 #2; p. 221 #1 (c); p. 223 #2; p. 225 #2; p. 227 #2. Note: the solutions to these questions begin on page 229.



Lesson 11: L’Hôpital’s Rule 1. Solve the following limits:

(a) 30

sinlimx

x xx→

− (b)

2

10lim

sinx

xx−→

(c)

( )( )

2lim sec cos3

xx x

π−→

(d) 0

lim lnx

x x+→

(e) 23lim

x

xx e→∞

− (f) ( )0

1limx

xxx e→

+

(g) 1

lim cosx

x

x→∞

(h) ( )0

cotlim 1 sin4x

xx+→

+

Homework for Lesson 11: Repeat question 1 in this lesson. Do the following questions from the old exams: p. 205 #3; p. 207 #2; p. 209 #4;

p. 211 #3; p. 213 #4; p. 215 #4; p. 217 #6; p. 220 #6; p. 221 #2; p. 223 #3; p. 225 #3; p. 227 #4. Note: the solutions to these questions begin on page 229.

Lesson 12: Improper Integrals and the Comparison Theorem 1. Determine if the following improper integrals converge or diverge. Evaluate those that

are convergent.

(a)

5

22

dxx

⌠⌡ −

(b) 2 3

3

sec x dxπ

π∫

(c) 21dx

x

∞

−∞

⌠⌡ +

(d) 3

01

dxx

⌠⌡ −



2. Use the Comparison Test to determine if the following integrals converge or diverge.

(a) 3

4

0

dxx x

⌠⌡ +

(b) 3

1

dxx x

⌠⌡

∞

+

(c) 2

2

1

sin xdx

x

⌠⌡

∞

(d)

1

1 xdx

x

⌠⌡

∞

+

(e) 1

2xdx

x e

⌠⌡

∞

+ (f)

1

0

xedx

x

⌠⌡

−

Homework for Lesson 12: Repeat questions 1 and 2 in this lesson. Do the following questions from the old exams: p. 205 #4; p. 207 #3; p. 209 #5;


Lesson 13: Arc Length and Surface Area

2

1dy

dL dxdx

= +

or 2

1dx

dL dydy

= +

If the curve is rotated about the x-axis: 2b

a

S y dLπ= ∫

If the curve is rotated about the y-axis: 2b

a

S x dLπ= ∫

1. Find the lengths of the following curves: (a) 2, 0 1y ex x= + ≤ ≤ (b) ( )ln sin , 6 3x y yπ π= ≤ ≤

(c) 3 1

, 1 26 2x

y xx

= + ≤ ≤ (d) 2 ln

, 2 42 4x x

y x= − ≤ ≤



2. Set up but do not solve the integrals to find the surface area if the curve 3 2 4y x x= + + on [0, 2] is revolved about:

(a) the x-axis (b) the y-axis (c) the x = 3 line (d) the y = –2 line 3. Find the surface area obtained by rotating the first quadrant region of the curve

21 , 1 5x y x= + ≤ ≤ about: (a) the x-axis (solve this integral) (b) the y-axis (set up the integral, but do not solve it) Homework for Lesson 13: Repeat questions 1 to 3 in this lesson. Do the following questions from the old exams: p. 206 #5 (a) and (b); p. 207 #4 (b)

and (e); p. 210 #6 (a) and (b); p. 211 #5 (c) and (d); p. 214 #6; p. 216 #6; p. 218 #8; p. 220 #5; p. 221 #4 (b); p. 224 #8; p. 226 #5 (c); p. 228 #7. Note: the solutions to these questions begin on page 229.

Lesson 14: Parametric Equations

2 2dx dydL dt

dt dt = +

( )1 2

x b

x a

Area y y dx=

=

= −∫ or ( )1 2

y b

y a

Area x x dy=

=

= −∫

1. For the parametric equations given below find dydx

and 2

2

d ydx

.

(a) 3cos , 5tanx t y t= = (b) 3 24 5, 4 6 9x t y t t= + = − + 2. Find the equation of the line tangent to the parametric curve lnx t= , ty te= at t = 1. 3. Find the equation of the line tangent to the parametric curve 2 3x t= + , 2 2y t t= + at

the point (5, 3). 4. Find the area bounded by the curve 1x t t= − , 1y t t= + and the line y = 5/2.



5. For the curve defined parametrically by ( )2 3x t t= − , ( )23 3y t= − , do the following:

(a) Show the point (0, 0) has 2 tangent lines. (b) Find the points where the curve has either a horizontal or a vertical tangent line. (c) Sketch the curve. (d) Set up but do not solve the integrals expressing: (i) The circumference of the enclosed region. (ii) The area of the enclosed region. (iii) The surface area of the solid created by rotating the enclosed region about

the x-axis. 6. Sketch the parametric curves defined below, by first doing the following: (i) Find the points which have vertical or horizontal tangent lines. (ii) Use the first derivative to establish the directional information of the curve. (iii) Use the second derivative to establish the concavity of the curve. (a) 2 22 , 4x t t y t t= − = − (b) 2 32 , 12x t y t t= = − [Hint: Verify that this curve has two tangents at (24, 0).] Homework for Lesson 14: Repeat questions 1 to 6 in this lesson. Do the following questions from the old exams: p. 206 #6; p. 208 #5; p. 210 #7;




Lesson 15: Polar Curves

22 dr

dL r dd

θθ

= +

212

b

a

A r dθ= ∫

1. Sketch the following polar curves: (a) r = 5 (b) 2sinr θ= (c) 4 cosr θ= − (d) ( )3 1 cosr θ= −

(e) 1 2sinr θ= + (f) sin2r θ= (g) 4 cos3r θ= (h) , 1r θ θ= ≥ 2. Find the area of one petal of the rose cos2r θ= . 3. Set up but do not solve the integral for the circumference of the cardioid 1 sinr θ= + .

4. Find the slope of the tangent line to 1 cosr θ= + at 6π

θ = .

Homework for Lesson 15: Repeat questions 1 to 4 in this lesson. Do the following questions from the old exams: p. 210 #8; p. 214 #8; p. 216 #8;

p. 218 #10; p. 220 #9; p. 222 #6; p. 226 #8; p. 228 #9. Note: the solutions to these questions begin on page 229.



SAMPLE







IT’S FREE!





tips to send.






services.


SAMPLE














made-to-order.



















SAMPLE



SAMPLE







IT’S FREE!





tips to send.






services.


SAMPLE














made-to-order.



















SAMPLE

calculus 2 - grant's tutoring · grant’s tutoring (text or call (204) 489-2884) do not...

Documents