calcii_complete_practice.pdf

7/28/2019 CalcII_Complete_Practice.pdf

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CALCULUS IIPractice Problems

Paul Dawkins


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Calculus II

Table of Contents

Preface ........................................................................................................................................... iiiOutline ........................................................................................................................................... iiiIntegration Techniques ................................................................................................................. 6

Introduction ................................................................................................................................................ 6Integration by Parts .................................................................................................................................... 6Integrals Involving Trig Functions ............................................................................................................. 7Trig Substitutions ....................................................................................................................................... 8Partial Fractions ......................................................................................................................................... 9Integrals Involving Roots ..........................................................................................................................10Integrals Involving Quadratics ..................................................................................................................10Integration Strategy ...................................................................................................................................11Improper Integrals .....................................................................................................................................11Comparison Test for Improper Integrals ...................................................................................................12Approximating Definite Integrals .............................................................................................................12

Applications of Integrals ............................................................................................................. 13Introduction ...............................................................................................................................................13Arc Length ................................................................................................................................................14Surface Area ..............................................................................................................................................14Center of Mass ..........................................................................................................................................15Hydrostatic Pressure and Force .................................................................................................................15Probability .................................................................................................................................................17

Parametric Equations and Polar Coordinates .......................................................................... 18Introduction ...............................................................................................................................................18Parametric Equations and Curves .............................................................................................................19Tangents with Parametric Equations .........................................................................................................19Area with Parametric Equations ......... ........... ........... .......... ........... .......... ........... .......... ........... .......... ........19Arc Length with Parametric Equations .....................................................................................................19Surface Area with Parametric Equations...................................................................................................19Polar Coordinates ......................................................................................................................................19Tangents with Polar Coordinates ..............................................................................................................19Area with Polar Coordinates .....................................................................................................................20

Arc Length with Polar Coordinates .......... .......... ........... .......... ........... .......... ........... .......... ........... ........... ..20Surface Area with Polar Coordinates ........................................................................................................20Arc Length and Surface Area Revisited .......... .......... ........... .......... ........... .......... ........... .......... ........... ......20

Sequences and Series ................................................................................................................... 20Introduction ...............................................................................................................................................20Sequences ..................................................................................................................................................21More on Sequences ...................................................................................................................................21Series The Basics ...................................................................................................................................21Series Convergence/Divergence ............................................................................................................21Series Special Series ..............................................................................................................................22Integral Test ..............................................................................................................................................22Comparison Test / Limit Comparison Test ...............................................................................................22Alternating Series Test ..............................................................................................................................22Absolute Convergence ..............................................................................................................................22Ratio Test ..................................................................................................................................................22Root Test ...................................................................................................................................................22Strategy for Series .....................................................................................................................................23Estimating the Value of a Series ...............................................................................................................23Power Series ..............................................................................................................................................23Power Series and Functions ......................................................................................................................23Taylor Series .............................................................................................................................................23Applications of Series ...............................................................................................................................23

2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

Binomial Series .........................................................................................................................................23Vectors .......................................................................................................................................... 23

Introduction ...............................................................................................................................................24Vectors The Basics ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... ........... ..24Vector Arithmetic .....................................................................................................................................24Dot Product ...............................................................................................................................................24Cross Product ............................................................................................................................................25

Three Dimensional Space............................................................................................................ 25

Introduction ...............................................................................................................................................25The 3-D Coordinate System ......................................................................................................................26Equations of Lines ....................................................................................................................................26Equations of Planes ...................................................................................................................................26Quadric Surfaces .......................................................................................................................................26Functions of Several Variables .................................................................................................................26Vector Functions .......................................................................................................................................26Calculus with Vector Functions ................................................................................................................26Tangent, Normal and Binormal Vectors ...................................................................................................26Arc Length with Vector Functions ............................................................................................................27Curvature...................................................................................................................................................27Velocity and Acceleration .........................................................................................................................27Cylindrical Coordinates ............................................................................................................................27Spherical Coordinates ...............................................................................................................................27

2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

Preface

Here are a set of practice problems for my Calculus II notes. If you are viewing the pdf versionof this document (as opposed to viewing it on the web) this document contains only the problems

themselves and no solutions are included in this document. Solutions can be found in a numberof places on the site.

1. If youd like a pdf document containing the solutions go to the note page for the sectionyoud like solutions for and select the download solutions link from there. Or,

2. Go to the download page for the site http://tutorial.math.lamar.edu/download.aspx andselect the section youd like solutions for and a link will be provided there.

3. If youd like to view the solutions on the web or solutions to an individual problem youcan go to the problem set web page, select the problem you want the solution for. At this

point I do not provide pdf versions of individual solutions, but for a particular problem

you can select Printable View from the Solution Pane Options to get a printableversion.

Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problems

although this will vary from section to section.

Outline

Here is a list of sections for which problems have been written.

Integration TechniquesIntegration by PartsIntegrals Involving Trig FunctionsTrig Substitutions

Partial FractionsIntegrals Involving RootsIntegrals Involving QuadraticsUsing Integral TablesIntegration Strategy

Improper IntegralsComparison Test for Improper IntegralsApproximating Definite Integrals

Applications of Integrals

Arc Length No problems written yet.Surface Area No problems written yet.

2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

Center of Mass No problems written yet.Hydrostatic Pressure and Force No problems written yet.

Probability No problems written yet.

Parametric Equations and Polar CoordinatesParametric Equations and Curves No problems written yet.

Tangents with Parametric Equations No problems written yet.Area with Parametric Equations No problems written yet.Arc Length with Parametric Equations No problems written yet.Surface Area with Parametric Equations No problems written yet.Polar Coordinates No problems written yet.Tangents with Polar Coordinates No problems written yet.

Area with Polar Coordinates No problems written yet.Arc Length with Polar Coordinates No problems written yet.Surface Area with Polar Coordinates No problems written yet.Arc Length and Surface Area Revisited No problems written yet.

Sequences and Series

Sequences No problems written yet.More on Sequences No problems written yet.Series The Basics No problems written yet.

Series Convergence/Divergence No problems written yet.Series Special Series No problems written yet.Integral Test No problems written yet.Comparison Test/Limit Comparison Test No problems written yet.Alternating Series Test No problems written yet.

Absolute Convergence No problems written yet.Ratio Test No problems written yet.Root Test No problems written yet.Strategy for Series No problems written yet.

Estimating the Value of a Series No problems written yet.Power Series No problems written yet.Power Series and Functions No problems written yet.Taylor Series No problems written yet.Applications of Series No problems written yet.Binomial Series No problems written yet.

VectorsVectors The Basics No problems written yet.Vector Arithmetic No problems written yet.Dot Product No problems written yet.

Cross Product No problems written yet.

Three Dimensional SpaceThe 3-D Coordinate System No problems written yet.Equations of Lines No problems written yet.

Equations of Planes No problems written yet.Quadric Surfaces No problems written yet.

Functions of Several Variables No problems written yet.Vector Functions No problems written yet.Calculus with Vector Functions No problems written yet.

2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

Tangent, Normal and Binormal Vectors No problems written yet.Arc Length with Vector Functions No problems written yet.

Curvature No problems written yet.Velocity and Acceleration No problems written yet.

Cylindrical Coordinates No problems written yet.Spherical Coordinates No problems written yet.

2007 Paul Dawkins v http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

Integration Techniques

Introduction

Here are a set of practice problems for the Integration Techniques chapter of my Calculus IInotes. If you are viewing the pdf version of this document (as opposed to viewing it on the web)this document contains only the problems themselves and no solutions are included in this

document. Solutions can be found in a number of places on the site.




point I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.

Note that some sections will have more problems than others and some will have more or less of

a variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.

Here is a list of topics in this chapter that have practice problems written for them.

Integration by PartsIntegrals Involving Trig Functions

Trig SubstitutionsPartial FractionsIntegrals Involving RootsIntegrals Involving QuadraticsUsing Integral Tables

Integration StrategyImproper IntegralsComparison Test for Improper IntegralsApproximating Definite Integrals

Integration by Parts

Evaluate each of the following integrals.

1. ( )4 cos 2 3x x dx

2007 Paul Dawkins 6 http://tutorial.math.lamar.edu/terms.aspx


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Calculus II

2. ( )13

0

6

2 5x

x dx+ e

3. ( ) ( )23 sin 2t t t dt +

4. ( )1 86 tan w dw

5. ( )2 14cosz

z dze

6. ( )20

cos 4x x dx

7. ( )7 4sin 2t t dt

8. ( )6 cos 3y y dy

9. ( )3 24 9 7 3 xx x x dx + + e

Integrals Involving Trig Functions


1. ( ) ( )3 42 23 3sin cosx x dx

2. ( ) ( )8 5sin 3 cos 3z z dz

3. ( )4cos 2t dt

4. ( ) ( )2

3 51 1

2 2cos sinw w dw

5. ( ) ( )6 2sec 3 tan 3y y dy

6. ( ) ( )3 10tan 6 sec 6x x dx

7. ( ) ( )4 7 30

tan secz z dz

8. ( ) ( )cos 3 sin 8t t dt



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Calculus II

9. ( ) ( )3

1

sin 8 sinx x dx

10. ( ) ( )4cot 10 csc 10z z dz

11. ( ) ( )6 41 14 4csc cotw w dw

12.( )

( )

4

9

sec 2

tan 2

tdt

t

13.( )

( )

3

2

2 7 sin

cos

zdz

z

+

14. ( ) ( ) ( )5 3 4

9sin 3 2cos 3 csc 3x x x dx

Trig Substitutions

For problems 1 8 use a trig substitution to eliminate the root.

1.2

4 9z

2.2

13 25x+

3. ( )5

227 3t

4. ( )2

3 100w +

5. ( )2

4 9 5 1t +

6.2

1 4 2z z

7. ( )3

228 21x x +

8.8

9x e

For problems 9 16 use a trig substitution to evaluate the given integral.



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Calculus II

9.

2

4

16xdx

x

+

10.

2

1 7w dw

11. ( )5

23 23 4t t dt

12.

5

4 27

2

25dy

y y

13.4

5 2

1

2 2 9z z dz+

14.2

1

9 36 37

dxx x +

15.( )

( )32

5

2

3

40 6

zdz

z z

+

16. ( ) ( )2cos 9 25sinx x dx+

Partial Fractions


1.2

4

5 14dx

x x+

2.2

8 3

10 13 3

tdt

t t

+

3.( ) ( )( )

02

1

7

2 1 4

w wdw

w w w

+

+

4.3 2

8

3 7 4dx

x x x+ +



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Calculus II

5.( )( )

42

2

2

3 1

1 5

zdz

z z

+

+

6.3 2

4 11

9

xdx

x x

7.( )( )

2

2

2 3

6 4

z zdz

z z

+ +

+

8.( )( )

2 3

2 2

8 6 12

3 4 7

t t tdt

t t

+ +

+ +

9.

( )( )

26 3

2 4

x xdx

x x

+

10.

4

3

2

9

wdw

w w

+

+

Integrals Involving Roots


1.7

2 4dx

x+

2.1

2 1 2dw

w w+ +

3.2

3 2 4 2

tdt

t t

+

Integrals Involving Quadratics


1.2

7

3 3dw

w w+ +



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Calculus II

2.2

10

4 8 9

xdx

x x +

3.

( )

522

2 9

14 46

tdt

t t

+

+

4.

( )2

2

3

1 4 2

zdz

z z

Integration Strategy

Problems have not yet been written for this section.

I was finding it very difficult to come up with a good mix of new problems and decided mytime was better spent writing problems for later sections rather than trying to come up with a

sufficient number of problems for what is essentially a review section. I intend to come back at alater date when I have more time to devote to this section and add problems then.

Improper Integrals

Determine if each of the following integrals converge or diverge. If the integral convergesdetermine its value.

1. ( )0

1 2xx dx

+ e

2. ( )0

1 2x

x dx

+ e

3.

1

5

1

10 2dz

z +

4.

2

3 21

4

4

w

dww

5.1

6 y dy

6.( )

4

2

9

1 3dz

z



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Calculus II

7.

4

2

0 9

xdx

x

8.( )

3

24

6

1

wdw

w

+

9.

4

2

1

1

6dx

x x+

10.

10

2

x

dxx

e

Comparison Test for Improper Integrals

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

1.3

1

1

1dx

x

+

2.

2

33 1

zdz

z

3.4

y

dyy

e

4.4 2

1

1

2

zdz

z z

+

5.( )( )

2

3 2

6

1

cos 1

wdw

w w

+

+

Approximating Definite Integrals

For each of the following integrals use the given value ofn to approximate the value of thedefinite integral using

(a) the Midpoint Rule,

(b) the Trapezoid Rule, and



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Calculus II

(c) Simpsons Rule.

Use at least 6 decimal places of accuracy for your work.

1.

7

3

1

1

1

dx

x +

using 6n =

2.22

1

1x

dx

+ e using 6n =

3. ( )4

0

cos 1 x dx+ using 8n =

Applications of Integrals

Introduction

Here are a set of practice problems for the Applications of Integrals chapter of my Calculus IInotes. If you are viewing the pdf version of this document (as opposed to viewing it on the web)this document contains only the problems themselves and no solutions are included in this

document. Solutions can be found in a number of places on the site.




point I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printableversion.

Note that some sections will have more problems than others and some will have more or less of

a variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.


Arc Length No problems written yet.Surface Area No problems written yet.

Center of Mass No problems written yet.Hydrostatic Pressure and Force No problems written yet.Probability No problems written yet.



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Calculus II

Arc Length

1. Set up, but do not evaluate, an integral for the length of 2y x= + , 1 7x using,

(a)

2

1dy

ds dxdx

= +

(b)

2

1dx

ds dydy

= +

2. Set up, but do not evaluate, an integral for the length of ( )cosx y= , 120 x using,

(a)

2

1dy

ds dxdx

= +

(b)

2

1dx

ds dydy

= +

3. Determine the length of ( )32

7 6y x= + , 189 875y .

4. Determine the length of ( )2

4 3x y= + , 1 4y .

Surface Area

1. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

5x y= + , 5 3x about the y-axis using,

(a)

2

1dy

ds dxdx

= +

(b)

2

1dx

ds dydy

= +

2. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating

( )sin 2y x= , 80 x about the x-axis using,

(a)

2

1dy

ds dxdx

= +



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Calculus II

(b)

2

1dx

ds dydy

= +

3. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating3

4y x= + ,

1 5x about the given axis. You can use eitherds.(a)x-axis

(b)y-axis

4. Find the surface area of the object obtained by rotating2

4 3y x= + , 1 2x about the y-axis.

5. Find the surface area of the object obtained by rotating ( )sin 2y x= , 80 x about the x-

axis.

Center of Mass

Find the center of mass for each of the following regions.

1. The region bounded by2

4y x= that is in the first quadrant.

2. The region bounded by 3xy = e , the x-axis, 2x = and they-axis.

3. The triangle with vertices (0, 0), (-4, 2) and (0,6).

Hydrostatic Pressure and Force

Find the hydrostatic force on the following plates submerged in water as shown in each image. In

each case consider the top of the blue box to be the surface of the water in which the plate issubmerged. Note as well that the dimensions in many of the images will not be perfectly to scalein order to better fit the plate in the image. The lengths given in each image are in meters.

1.



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Calculus II

2.

3.



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Calculus II

Probability

1. Let,

( )( )2

320 if 2 18

37760

0 otherwise

x x xf x

=

(a) Show that( )

f x is a probability density function.

(b) Find ( )7P X .(c) Find ( )7P X .(d) Find ( )3 14P X .(e) Determine the mean value ofX.

2. For a brand of light bulb the probability density function of the life span of the light bulb isgiven by the function below, where tis in months.

( )250.04 if 0

0 if 0

t

tf t

t

=


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Calculus II

3. Determine the value ofc for which the function below will be a probability density function.

( )( )3 48 if 0 8

0 otherwise

c x x xf x

=

Parametric Equations and Polar Coordinates

Introduction

Here are a set of practice problems for the Parametric Equations and Polar Coordinates chapter ofmy Calculus II notes. If you are viewing the pdf version of this document (as opposed to viewing

it on the web) this document contains only the problems themselves and no solutions are includedin this document. Solutions can be found in a number of places on the site.

10.If youd like a pdf document containing the solutions go to the note page for the sectionyoud like solutions for and select the download solutions link from there. Or,

11.Go to the download page for the site http://tutorial.math.lamar.edu/download.aspx andselect the section youd like solutions for and a link will be provided there.

12.If youd like to view the solutions on the web or solutions to an individual problem youcan go to the problem set web page, select the problem you want the solution for. At this






Parametric Equations and Curves No problems written yet.

Tangents with Parametric Equations No problems written yet.Area with Parametric Equations No problems written yet.

Arc Length with Parametric Equations No problems written yet.Surface Area with Parametric Equations No problems written yet.Polar Coordinates No problems written yet.

Tangents with Polar Coordinates No problems written yet.Area with Polar Coordinates No problems written yet.Arc Length with Polar Coordinates No problems written yet.Surface Area with Polar Coordinates No problems written yet.Arc Length and Surface Area Revisited No problems written yet.



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Calculus II

Parametric Equations and Curves


Tangents with Parametric Equations


Area with Parametric Equations


Arc Length with Parametric Equations


Surface Area with Parametric Equations


Polar Coordinates


Tangents with Polar Coordinates




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Calculus II

Area with Polar Coordinates


Arc Length with Polar Coordinates


Surface Area with Polar Coordinates


Arc Length and Surface Area Revisited

Problems have not yet been written for this section and probably wont be to be honest since thisis just a summary section.

Sequences and Series

Introduction

Here are a set of practice problems for the Sequences and Series chapter of my Calculus II notes.If you are viewing the pdf version of this document (as opposed to viewing it on the web) thisdocument contains only the problems themselves and no solutions are included in this document.Solutions can be found in a number of places on the site.


14.Go to the download page for the site http://tutorial.math.lamar.edu/download.aspx andselect the section youd like solutions for and a link will be provided there.15.If youd like to view the solutions on the web or solutions to an individual problem you

can go to the problem set web page, select the problem you want the solution for. At thispoint I do not provide pdf versions of individual solutions, but for a particular problem




22/28

Calculus II




Sequences No problems written yet.More on Sequences No problems written yet.Series The Basics No problems written yet.Series Convergence/Divergence No problems written yet.Series Special Series No problems written yet.Integral Test No problems written yet.

Comparison Test/Limit Comparison Test No problems written yet.Alternating Series Test No problems written yet.Absolute Convergence No problems written yet.Ratio Test No problems written yet.

Root Test No problems written yet.Strategy for Series No problems written yet.

Estimating the Value of a Series No problems written yet.Power Series No problems written yet.Power Series and Functions No problems written yet.

Taylor Series No problems written yet.Applications of Series No problems written yet.Binomial Series No problems written yet.

Sequences


More on Sequences


Series The Basics


Series Convergence/Divergence




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Calculus II

Series Special Series


Integral Test


Comparison Test / Limit Comparison Test


Alternating Series Test


Absolute Convergence


Ratio Test


Root Test




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Calculus II

Strategy for Series


Estimating the Value of a Series


Power Series


Power Series and Functions


Taylor Series


Applications of Series


Binomial Series


Vectors



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Calculus II

Introduction

Here are a set of practice problems for the Vectors chapter of my Calculus II notes. If you areviewing the pdf version of this document (as opposed to viewing it on the web) this documentcontains only the problems themselves and no solutions are included in this document. Solutions

can be found in a number of places on the site.




point I do not provide pdf versions of individual solutions, but for a particular problemyou can select Printable View from the Solution Pane Options to get a printable

version.

Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.


Vectors The Basics No problems written yet.Vector Arithmetic No problems written yet.

Dot Product No problems written yet.Cross Product No problems written yet.

Vectors The Basics


Vector Arithmetic


Dot Product




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Calculus II

Cross Product


Three Dimensional Space

Introduction

Here are a set of practice problems for the Three Dimensional Space chapter of my Calculus II

notes. If you are viewing the pdf version of this document (as opposed to viewing it on the web)this document contains only the problems themselves and no solutions are included in thisdocument. Solutions can be found in a number of places on the site.









The 3-D Coordinate System No problems written yet.Equations of Lines No problems written yet.Equations of Planes No problems written yet.

Quadric Surfaces No problems written yet.Functions of Several Variables No problems written yet.Vector Functions No problems written yet.

Calculus with Vector Functions No problems written yet.Tangent, Normal and Binormal Vectors No problems written yet.

Arc Length with Vector Functions No problems written yet.Curvature No problems written yet.Velocity and Acceleration No problems written yet.Cylindrical Coordinates No problems written yet.Spherical Coordinates No problems written yet



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Calculus II

The 3-D Coordinate System


Equations of Lines


Equations of Planes


Quadric Surfaces


Functions of Several Variables


Vector Functions


Calculus with Vector Functions


Tangent, Normal and Binormal Vectors




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Calculus II

Arc Length with Vector Functions


Curvature


Velocity and Acceleration


Cylindrical Coordinates


Spherical Coordinates


calcii_complete_practice.pdf

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