Math 172 Chapter 9A notes
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9.1 Parametric Curves
So far we have discussed equations in the form
a parameter.
. Sometimes
Example. Projectile Motion
Sketch
and
axes, cannon at origin, trajectory
Mechanics gives
and
. Time is a parameter. ■
Given parameter . Then
,
are parametric equations for a curve in the
Example.
,
Draw the curve in the
t
0
1
2
-plane.
x
2
1
0
-plane.
y
3
1
-1
Sketch axes and line in
-plane
Eliminate
■
Example
where
and
are given as functions of
Math 172 Chapter 9A notes
-,
-, plot points, draw portion of circle with arrow, show
Parameter is the angle .
Eliminate
,
Gives the first quadrant portion of a circle of radius . ■
Example.
hyperbola with asymptotes
sketch
-axes, asymptotes, hyperbola
parametric equations describe the top branch of the hyperbola ■
A cycloid is a curve traced by a point on the rim of a rolling wheel.
sketch wheel, wheel rolled about a quarter turn ahead, portion of cycloid
Find parametric equations
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Math 172 Chapter 9A notes
circle has radius
a point on the cycloid
length of arc
Parametric equations for cycloid
Table
0
sketch
-axes, plot points, draw curve through them
one arch of the cycloid
Drawing Graphs of Parametric Equations using Maple
Command form: plot([ -expression, -expression, parameter range],scaling=constrained);
Show graph of parametric equations
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Math 172 Chapter 9A notes
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9.2 Calculus with parametric curves
Tangents
Curve
in
-plane described by parametric equations
Chain rule
This gives the slope of curve
Let
This gives the concavity of the curve
Example
(a) Find the equation of the line tangent to the curve at
At
equation of tangent line
.
Math 172 Chapter 9A notes
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or
(b) Sketch the curve and the tangent line
sketch
(c) Find
-axes, asymptotes, plot points, draw upper branch of hyperbola, sketch tangent line
and discuss the concavity of the curve.
Since
, for
The top branch of the hyperbola is concave up.
Example. Cycloid
,
Table with additional row
Math 172 Chapter 9A notes
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0
-,
(a) Discuss the slope at
-, plot points, draw cycloid
and in the limit as
At
add segment with slope 0
At
add segment with slope 1
As
where L’Hospital’s rule has been used.
The cycloid has a vertical tangent in the limit
(b) Discuss concavity of the cycloid
.
add
Math 172 Chapter 9A notes
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for
The cycloid is concave down over the entire arch, except for the cusp points
defined. ■
Areas
sketch
-,
, curve
above
Area under the curve
Suppose
, where
and
Example. Find the area of the circle
,
sketch
,
-axes, unit circle, angle
CCW from positive -axis
Notice the last integral integrates over a full period of cosine. ■
Example. Find the area of the asteroid
where it is not
Math 172 Chapter 9A notes
sketch
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-,
-, astroid
therefore
so
Notice
The first two integrals are seen to be zero by symmetry because the integrands are odd powers of
cosine and the argument varies over a full period.
The value of the last integral can be seen from the fact that the average value of sin2 or cos2 over their
period
is ½. It is also an immediate consequence of the half angle identities.
■
Math 172 Chapter 9A notes
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Arc Length
Symbolically
-,
Suppose
-, curve
over
, triangle
is described by parametric equations
then
where
and
.
Example Find the length of the curve
■
Example Find the total length of the astroid
Math 172 Chapter 9A notes
-,
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-, astroid
For
and
Therefore
where we used
■
9.3 Polar Coordinates
Cartesian or rectangular coordinates
is associated with a unique ordered pair
Math 172 Chapter 9A notes
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Polar Coordinates
…0…, …0…, indicate the polar axis, ray,
distance from origin
angle measured CCW from the polar axis
Example. Plot
■
ray,
Representation in polar coordinates is not unique.
axes,
, indicate backward extension of ray through origin
may be represented by
,
,
Example. May represent
by
■
Sometimes restrict
every point except
has a unique representation.
anything
always represents the origin
Conversion from polar to Cartesian coordinates
,
ray,
, projection from tip to -axis
Math 172 Chapter 9A notes
Example. Convert
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to Cartesian coordinates
■
Conversion from Cartesian to polar coordinates
Example. Convert
to polar coordinates with
and
.
,
[1a]
[1b]
Solutions of [1] with
and
:
,
Notice that
But
is not the same point as
Equations [1] are not sufficient, we must also choose
to be in the correct quadrant. ■
Math 172 Chapter 9A notes
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Polar Equations
General form
Common form
Example.
axes, circle of radius
circle, center at origin, with radius
To find equation in Cartesian coordinates, square both sides:
■
giving
Example. Find the polar equation for the curve represented by
[2]
Let
and
, then
Eq. [2] becomes
Solutions are
or
[2] is an equation for a circle. To see, complete squares
sketch axes, circle centered at
circle with radius
and center
with radius
. ■
Math 172 Chapter 9A notes
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Symmetry of solutions of
graph axes,
and ray in first quadrant,
(1) If
whenever
and ray,
and ray,
, the solution is symmetric about the -axis.
(2) If
whenever
, the solution is symmetric about the -axis.
(3) If
whenever
, the solution is symmetric about the origin.
Example.
and ray
.
.
Solution is symmetric about the -axis. ■
Example.
Solution is symmetric about the -axis
Sketch
sketch points in first quadrant, draw smooth curve through them, complete in fourth quadrant
This is a cardioid. ■
Tangents to Polar Curves
Common form of a polar equation
Math 172 Chapter 9A notes
where
Consider
and
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.
as a parameter, then from the results of section 9.2
Let
Suppose the graph of
axes, ray
passes through the origin at an angle
, curve passing through origin tangent to ray
slope =
add projection down from ray to -axis to complete a right triangle
Example. Find the slope of the line tangent to
at
.
At
,
,
Thus
,
,
Math 172 Chapter 9A notes
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Sketch
sketch axes, ray
, plot points, draw curve
9.4 Areas and Lengths in Polar
Coordinates
Area of a sector of a circle
circle of radius , sector subtended by , area
Area bounded by a polar curve
dashed polar axis, ray
subtended by , its angle
Area of slice with angle
Total area bounded by
:
, ray
, curve
:
and the rays
Example. Find the area enclosed by the loop of
Use the half-angle identity
between these angles, small sector
and
between
and
.
Math 172 Chapter 9A notes
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. ■
with the substitution
Example. Find the area of the region that lies inside the graph of
but outside the graph of
plot points for cardioids and draw curve (lower half plane by
symmetry), plot points for circle and draw curve. Area has components in the first and second
quadrants.
Area between curves is
.
Find intersections between curves
This misses the intersection at the origin! Why? Graphs have different values of
at the origin.
Math 172 Chapter 9A notes
Then
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. ■
. Area between the curves
Arc Lengths in Polar Coordinates
sketch
, ray
, ray
Symbolically
where
Keeping in mind that depends on
, curve
Math 172 Chapter 9A notes
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Thus
Example. Find the length of the cardioid
where
consider
for
Then
.
Math 172 Chapter 9A notes
■
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