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Where is it? Coordinate systems are used to locate the position of a point. (3,1) (1,/6) 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. 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.
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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