ma132 final exam review. 6.1 area between curves partition into rectangles! area of a rectangle is a...
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![Page 1: MA132 Final exam Review. 6.1 Area between curves Partition into rectangles! Area of a rectangle is A = height*base Add those up ! (Think: Reimann Sum)](https://reader033.vdocument.in/reader033/viewer/2022051516/56649d565503460f94a337fd/html5/thumbnails/1.jpg)
MA132 Final exam
Review
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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”
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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
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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)
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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.
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ro ri
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Idea works for functions of y, too
y
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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
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Setting up the integralAnother cool movie
x
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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)
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Chapter 7: Techniques of Integration
• Integration by Parts
• Trig Integrals (i.e. using identities for clever u-sub)
• Trig Substitution
• Partial Fractions
• Improper Integrals
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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(
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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’
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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!
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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
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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)
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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!
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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
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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
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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
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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
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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
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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
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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
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9.3 Separable DEs
Integrate!
)()(
)()( Separate
Given
dxxgyh
dy
yhxgdx
dy
f(x,y)dx
dy
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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
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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
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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
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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!
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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
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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
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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
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Functions as Power Series
.xx
-x
n
n 1for
wasseries original themeanswhich
series geometric a of SUM theas 1
1 View
0
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Taylor and MacLaurin Series
• KNOW the MacLaurin series for – sin(x)– cos(x) – ex
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