ma132 final exam review. 6.1 area between curves partition into rectangles! area of a rectangle is a...
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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
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