10.3 Polar Coordinates. One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions.

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Presentation transcript:

10.3 Polar Coordinates

One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. r – the directed distance from the origin to a point Ө – the directed angle from the initial ray (x-axis) to ray OP.

(Circle centered at the origin) (Line through the origin) Some curves are easier to describe with polar coordinates: (Ex.: r = 2 is a circle of radius 2 centered around the origin) (Ex. Ө = π /3 is a line 60 degrees above the x-axis extending in both directions)

More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are: Each point can be coordinatized by an infinite number of polar ordered pairs.

Tests for Symmetry: x-axis: If (r,  ) is on the graph,so is (r, -  ).

Tests for Symmetry: y-axis: If (r,  ) is on the graph,so is (r,  -  )or (-r, -  ).

Tests for Symmetry: origin: If (r,  ) is on the graph,so is (-r,  )or (r,  +  ).

Tests for Symmetry: If a graph has two symmetries, then it has all three:

Try graphing this. (Pol mode)

Remember from trig, in polar coordinates, x = r cos Θ y = r sinΘ

To find the slope of a polar curve: We use the product rule here. A lot like parametric slope.

Example:

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:

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

Example: Find the area enclosed by: This graph is called a lima ƈon.

Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

When finding area, negative values of r cancel out: Area of one leaf times 4:Area of four leaves:

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

There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: