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Polar Coordinates

- Definition, Conversions, and Integration

Where is it?

Coordinate systems are used to locate the

position of a point.

- 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.

Locating points in Polar Coordinates

The first coordinate, r 2, indicates the

distance of the point from the origin.

The second coordinate, ? ?/6, indicates the

distance counter-clockwise around from the

positive x-axis.

Locating points in Polar Coordinates

Note, however, that every point in the plane as

infinitely many polar representations.

Locating points in Polar Coordinates

Note, however, that every point in the plane as

infinitely many polar representations.

Locating points in Polar Coordinates

Note, however, that every point in the plane as

infinitely many polar representations.

And we can go clockwise or counterclockwise

around the circle as many times as we wish!

Converting Between Polar and Rectangular

Coordinates

It is fairly easy to see that if (x,y) and (r, q)

represent the same point in the plane

These relationships allow us to convert back and

forth between rectangular and polar coordinates

Integration in Polar Coordinates

- Non-rectangular Integration Elements

Small Changes in r and q

Suppose we consider a small change from r . . .

to r dr

Small Changes in r and q

Suppose we consider a small change from r . . .

to r dr This gives us a thin ring around

the origin.

Small Changes in r and q

Suppose we consider a small change from q . . .

to q dq

Small Changes in r and q

Suppose we consider a small change from q . . .

to q dq This gives us a pie-shaped wedge

that is subtended by the angle dq.

Small Changes in r and q

Intersecting the thin ring . . . and the

pie-shaped wedge, . . . we get . . .

Small Changes in r and q

Intersecting the thin ring . . . and the

pie-shaped wedge, . . . we get . . .

Small Changes in r and q

In order to integrate a function given in polar

coordinates (without first converting to

rectangular coordinates!), we need to know the

area of this little piece.

dr

Why?

dq

Integration in Polar Coordinates

In order to integrate a function given in polar

coordinates, we will first chop up our region

into a bunch of concentric circles and rays

emanating from the origin.

(r,q, f(r,q))

(r,q)

Now do this for each little wedge and add up

the volumes of the towers.

Problem the volume of the tower is the area

of the base times the height. But the base is

not a rectangle, so its area is not dr dq!

Area of the Small Bit

A area of sector of a circle

q

Area of a Small Bit

In order to integrate a function given in polar

coordinates (without first converting to

rectangular coordinates!), we need to know the

area of this little piece.

dr

r

dq

Area of a Small Bit

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