MA132 Final exam Review

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MA132 Final examReview
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6.1 Area between curves
Partition into rectangles! Area of a rectangle is
A heightbase Add those up! (Think Reimann
Sum)
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 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)

• Consider a slice perpendicular to the axis of
  rotation

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Disks and Washer
ri
ro
ro is the distance from the line of rotation to
the outer curve. ri is the distance from the
line of rotation to the inner curve.
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ro
ri
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Idea works for functions of y, too
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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
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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

d
Radius here is just x
xd
x
x
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

• 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

Know it!
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7.2 Trig Integrals

• 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
Use Algebra to rewrite in this form
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Trig sub pitfalls

• Do NOT use the same variable when you make a
  change of variables
• EX. Let xsin(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
These are equal! We just need algebra!

• IDEA We do not know how to integrate
• But we do know how to integrate

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Undo the process of getting a common denominator

• Must be proper rational function
• Degree of numerator lt 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 eachwith powers up to the repeated
  value

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Forming the PFD depends on the factored Q(x)

• Q(x) includes irreducible quadratics
• Q(x) includes repeated irreducible quadratics

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Forming the PFD depends on the factored Q(x)

• Or a combination of all those!
• Example

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7.8 Improper Integrals
Integrating to infinity

• Two Types
• Infinite bounds
• Singularity between the bounds

Singularity at xa
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Plan of attack
These involve Integration AND limits

• 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

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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
• Newtons law of cooling

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9.3 Separable DEs
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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
These are separable differential equations

• 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

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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
Dont forget everything you know about
limits! Only apply LHopitals rule to
continuous functions of x Do NOT apply
series tests!
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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

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Power Series
For what values of x does the series converge
Make repeated use of the ratio test!
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Idea
We set Llt1 because That is when the Ratio Test
yields convergence

• Given
• Apply ratio test

This limit should include x-a Unless the limit
is 0 or infinity
Then use algebra to express This as x-altr
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Functions as Power Series
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Taylor and MacLaurin Series

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

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