10. 4 Polar Coordinates and Polar Graphs 10 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates

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.

Locating points in Polar Coordinates Suppose we see the point and we know it is in polar coordinates. Where is it in the plane? (r, )= (2,/6) r =2 (2,/6) The first coordinate, r =2, indicates the distance of the point from the origin.  = /6 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. (2,/6)  = /6

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

Graphing a Polar Equation Note that some points are “retraced” as you mark them.

Summary of Special Polar Graphs Limacons: Limacon with inner loop Cardiod (heart-shaped) Dimpled Limacon Convex Limacon

Rose Curves:

Circles and Lemniscates Note that a is the length of the diameter of the circles and length of the loop of a lemnicscate.

Some Calculus of Polar Curves To find the slope of a polar curve: We use the product rule here.

Example:

Example: Where are the horizontal and vertical tangents of

Area Inside a Polar Graph: The length of an arc (in a circle) is given by r. q when q is given in radians. For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:

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

Example: Find the area enclosed by:

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

Notes: To find the area between curves, subtract: 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

To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

Example: Find the length of the arc for the cardioid