9.1 parametric curves - washington state university-, -, plot points, draw portion of circle with...

Math 172 Chapter 9A notes of 20

9.1 Parametric Curves

So far we have discussed equations in the form . Sometimes and are given as functions of

a parameter.

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 -plane.

Example. ,

Draw the curve in the -plane.

t x y 0 2 3 1 1 1 2 0 -1

Sketch axes and line in -plane

Eliminate

■

Example where


-, -, 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


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


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


or

(b) Sketch the curve and the tangent line

sketch -axes, asymptotes, plot points, draw upper branch of hyperbola, sketch tangent line

(c) Find and discuss the concavity of the curve.

Since

, for

The top branch of the hyperbola is concave up.

Example. Cycloid

,

Table with additional row


0

-, -, plot points, draw cycloid

(a) Discuss the slope at 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 . add

(b) Discuss concavity of the cycloid


for

The cycloid is concave down over the entire arch, except for the cusp points where it is not

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


sketch -, -, 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.

■


Arc Length Symbolically

-, -, curve over , triangle

Suppose is described by parametric equations

then

where and .

Example Find the length of the curve

■

Example Find the total length of the astroid


-, -, astroid

For

and

Therefore

where we used

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9.3 Polar Coordinates

Cartesian or rectangular coordinates

is associated with a unique ordered pair


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


Example. Convert 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. ■


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 with radius

circle with radius and center . ■


Symmetry of solutions of graph axes, and ray in first quadrant, and ray, and ray, and ray

(1) If whenever , 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. .

.

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


where and .

Consider as a parameter, then from the results of section 9.2

Let

Suppose the graph of passes through the origin at an angle

axes, ray , 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


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 , ray , curve between these angles, small sector

subtended by , its angle

Area of slice with angle :

Total area bounded by and the rays and

Example. Find the area enclosed by the loop of between and .

Use the half-angle identity


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.


Then . Area between the curves . ■

Arc Lengths in Polar Coordinates sketch , ray , ray , curve

Symbolically

where

Keeping in mind that depends on


Thus

Example. Find the length of the cardioid .

where

consider

for

Then


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9.1 parametric curves - washington state university-, -, plot points, draw portion of circle with...

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