10.4 Polar Coordinates and Polar Graphs10.5 Area and Arc Length in Polar Coordinates

Where is it?

In rectangular coordinates: •We break up the plane into a grid of horizontal and vertical line lines. •We locate a point by identifying it as the intersection of a vertical and a horizontal line.

In polar coordinates:•We break up the plane with circles centered at the origin and with rays emanating from the origin.•We locate a point as the intersection of a circle and a ray.

Coordinate systems are used to locate the position of a point.

(3,1) (1,/6)

Locating points in Polar CoordinatesSuppose we see the point

and we know it is in polar coordinates. Where is it in the plane?

(r, )= (2,/6)

The first coordinate, r =2, indicates the distance of the point from the origin.

(2,/6)

The second coordinate, = /6, indicates the distance counter-clockwise around from the positive x-axis.

r =2

= /6

Locating points in Polar CoordinatesNote, however, that every point in the plane as infinitely many polar representations.

(2,/6)

= /6

( , ) 2, 6r

Locating points in Polar CoordinatesNote, however, that every point in the plane as infinitely many polar representations.

( , ) 2, 6r

2, 26

132, 6

2 6

Locating points in Polar CoordinatesNote, however, that every point in the plane as infinitely many polar representations.

( , ) 2, 6r

2, 26

112, 6

26

2, 26

And we can go clockwise or counterclockwise around the circle as many times as we wish!

Converting Between Polar and Rectangular Coordinates

,r

2 2 2

cos( )sin( )

tan( )

x ry r

r x yxy

It is fairly easy to see that if (x,y) and (r, ) represent the same point in the plane:

These relationships allow us to convert back and forth between

rectangular and polar coordinates

Graphing a Polar Equation4 sinr

x

y

Summary of Special Polar Graphs

1ab

Limacon with inner loop

1ab

Limacons:

Cardiod(heart-shaped)

DimpledLimacon

1 2ab

2ab

Convex Limacon

cossin

0, 0

r a br a ba b

Rose Curves:

cosr a b

petals if is odd2 petals if is even

2

b bb bb

cosr a b sinr a b sinr a b

Circles and Lemniscates

cosr a sinr a 2 2 sin 2r a 2 2 cos 2r a

Circles Lemniscates

To find the slope of a polar curve:

dydy d

dxdxd

sin

cos

d rdd rd

sin coscos sin

r rr r

We use the product rule here.

Some Calculus of Polar Curves

Example: 1 cosr

Example:Where are the horizontal and vertical tangents of sinr

The length of an arc (in a circle) is given by r. when is given in radians.

Area Inside a Polar Graph:

For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: 1

2A bh

r dr

21 1 2 2

dA rd r r d

We can use this to find the area inside a polar graph.

212

dA r d

21 2

dA r d

212

A r d

Example: Find the area enclosed by: 2 1 cosr

Example:Find the area of one petal of the rose curve given by

3cos3r

x

y

Notes:

To find the area between curves, subtract:

2 212

A R r d

Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

Example:Find the area of the region lying between the inner and outer loops of the

limacon sin21 r

To find the length of a curve:

Remember: 2 2ds dx dy

For polar graphs: cos sinx r y r

If we find derivatives and plug them into the formula, we (eventually) get:

22 drds r d

d

So: 22Length drr d

d

Example:Find the length of the arc for the cardioid

cos22 fr

x

y