MA132 Final exam

Review

6.1 Area between curves

n

iii xxgxf

1

)()(

b

a

ii dxxgxf )()(

Partition into rectangles!Area of a rectangle is A = height*baseAdd those up!(Think: Reimann Sum)

n

iii xxgxf

1

)()(

For the height, think “top – bottom”

6.2 Volumes by slicing• Given a region bounded

by curves

• Rotate that region about the x-axis, y-axis, or a horizontal or vertical line

• Generate a solid of revolution

• Partition into disks

6.2 Volume by slicing

• Consider a slice perpendicular to the axis of rotation

x

• Consider a slice perpendicular to the line of rotation•Label the thickness•This slice will be a disk or a washer•We can find the volume of those!•Consider a partition and add them up•(Think Reimann sum)

Disks and Washer

xrV 2

xrV 2 xrxrV io 22

rori

ro is the distance from the line of rotation to theouter curve. ri is the distance from the line of rotationto the inner curve.

ro ri

Idea works for functions of y, too

y

6.3 Volume by Shells

• Consider a rectangle parallel to the line of rotation

• Label the thickness• Rotating that rectangle around leads to a

cylindrical shell • We can find the volume of those!• Consider a partition and add them up• (Think Reimann sum)• A cool movie

Setting up the integralAnother cool movie

x

Shell Hints

• Draw the reference rectangle and a shell

• Label everything!

• The radius is just the distance from the line of rotation to the ‘reference rectangle’

• ALWAYS think in terms of distances

Radius here is just xx=d

d

xx

Radius here is (d – x)

Chapter 7: Techniques of Integration

• Integration by Parts

• Trig Integrals (i.e. using identities for clever u-sub)

• Trig Substitution

• Partial Fractions

• Improper Integrals

7.1 Parts: for handling products of functions

vduuvudv•Choose u so that differentiating leads to an easier function

•Choose dv so that you know how to integrate it!

•Be aware of boomerangs in life (not on the final)

•Careful:

cx

dxx

cxxxdxx

1

)ln(

)ln()ln(

7.2 Trig Integrals

(x)(x)x)(

xx

xx

(x)(x)

(x)(x)

cossin22sin

sinefor angle Double

)2cos(12

1)(sin

)2cos(12

1)(cos

Identities ReducingPower

sec1tan

1cossin

Identitiesn Pythagorea

2

2

22

22

dxxx

dxxx

mn

mn

)(sin)(cos

)(sec)(tan

• Use a trig identity to find an integral with a clever u-substituion!•Examine what the possibilities for ‘du’ are and then use the identities to get everything else in terms of ‘u’

7.4 Trig Substitution

• Use Pythagorean Identities

• Use a change of variables

• Rewrite everything in terms of trig functions– May have to apply more trig identities

• Change back to original variable!– May need to draw a right triangle!

7.3 Trig Sub

)tan(1)(sec

cos)(sin1

sec1tan

Identitiesn Pythagorea

2

22

22

xx

(x)x

(x)(x)

dxaxb

dxxba

dxaxb

222

222

222

Use Algebra to rewrite

in this form

)tan( ,)tan()sec( ),sec(

)cos( ,)cos( ),sin(

)sec( ,)(sec ),tan(

2222

2

2

2

2222

2

2

2

22222

2

2

2

axbadb

adx

b

ax

aaxbdb

adx

b

ax

aaxbdb

adx

b

ax

Trig sub pitfalls

• Do NOT use the same variable when you make a ‘change of variables’– EX. Let x=sin(x)

• Do NOT forget to include ‘dx’ when you rewrite your integral

• Do NOT forget to change BACK to the original variable– May involve setting up a right triangle– You may need to use sin(2x)=2sin(x)cos(x)

7.4 Partial Fractions

IDEA: We do not know how to integrate

But we do know how to integrate

dxxx

x

2

52

dx

xx 2

1

1

2These are equal!

We just need algebra!

Undo the process of getting a common denominator

dxQ(x)

P(x)Given

•Must be proper rational function

Degree of numerator < degree of denominator

FACTOR

product of linear terms and irreducible quadratic terms

FORM

FIND

Forming the PFD: depends on the factored Q(x)

• Q(x) includes distinct linear terms, include one of these for each one!

• Q(x) includes some repeated linear terms, include one term for each—with powers up to the repeated value

bax

A

32 )(,

)(,

bax

C

bax

B

bax

A

Forming the PFD: depends on the factored Q(x)

• Q(x) includes irreducible quadratics

• Q(x) includes repeated irreducible quadratics

cx

BAx

2

32222 )(,

)(,

cx

FEx

cx

DCx

cx

BAx

Forming the PFD: depends on the factored Q(x)

• Or a combination of all those!

Example:

2222

222

)1()1()1()1()2(

)1()1)(2(

42

x

GFx

x

EDx

x

C

x

B

x

A

xxx

x

7.8 Improper Integrals

Two Types:• Infinite bounds

• Singularity between the bounds

dxxf )(

b][a,in point someat y singularit

,)(b

adxxf

Singularity at x=a

Integrating to infinity

Plan of attack

• Rewrite using a dummy variable and in terms of a limit

• Integrate!

• Evaluate the limit of the result

• Analyze the result– A finite number: integral converges– Otherwise: integral diverges

These involve Integration

ANDlimits

Differential Equations

• An equation involving an unknown function and some of its derivatives

• We looked at separation of variables (9.3)

• Applications (9.4)– Growth/population models– Newton’s law of cooling

9.3 Separable DEs

Integrate!

)()(

)()( Separate

Given

dxxgyh

dy

yhxgdx

dy

f(x,y)dx

dy

Separable DEs

• Remember the constant of integration

• Initial value problems– Given an initial condition y(x0)=y0– Use to define the value of C

• Implicit solution vs. Explicit solution

9.4 Applications

• The rate of growth is proportional to the population size

• The rate of cooling is proportional to the temperature difference between the object and its surroundings

0)0(, PPkPdt

dP

0)0(),( TTTTkdt

dTs

These are separabledifferentialequations

Sequences and Series

• 11.1 Sequences

• 11.2 Series

• 11.4-11.6 Series tests (no 11.3)

• 11.8 Power series

• 11.9 functions of power series

• 11.10 MacLaurin and Taylor series

11.1 Sequences Some ideas

exists lim if converges nn

n aa

Don’t forget everythingyou know about limits!

Only apply L’Hopital’s rule to continuous functions of x

Do NOT apply seriestests!

Series

• Know which tests apply to positive series and ALL conditions for each test

• Absolute convergence means converges

• Absolute convergence implies convergence

• Conditional convergence means

converges BUT does NOT

0nna

0nna

0nna

Power Series

0

)(n

nn axc

Rax

Rax

Rx

for converges Series

0 i.e. ,for only converges Series

i.e. , allfor converges Series

Make repeated use of the ratio test!

For what values of x does the series converge

Idea

• Given

• Apply ratio test:

0

)(n

nn axc

1)(

)(lim

11

L

axc

axcn

n

nn

n

This limit should include|x-a|

Unless the limit is 0 or infinity

We set L<1 becauseThat is when the Ratio

Test yields convergence

Then use algebra to express

This as |x-a|<r

Functions as Power Series

.xx

-x

n

n 1for

wasseries original themeanswhich

series geometric a of SUM theas 1

1 View

0

Taylor and MacLaurin Series

• KNOW the MacLaurin series for – sin(x)– cos(x) – ex