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10.4 – 10.5 Polar Coordinates, Polar Graphs and Calculus

10.4 – 10.5 Polar Coordinates, Polar Graphs and Calculus. Pola r 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.”.

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10.4 – 10.5 Polar Coordinates, Polar Graphs and Calculus

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  1. 10.4 – 10.5 Polar Coordinates, Polar Graphs and Calculus

  2. 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. A polar coordinate pair Pole (origin) Initial ray determines the location of a point. Directed distance, r can be negative Directed angle

  3. Coordinate Conversion Note: Polar coordinates are not unique. Polar-to-Rectangular conversion: Rectangular-to-Polar conversion:

  4. Polar Graphs Describe the graph of each polar equation. Confirm it by converting to a rectangular equation. Some curves are easier to describe with polar coordinates: (Circle centered at the origin with radius a) (Line through the origin )

  5. Examples Convert to Rectangular Form. Convert to Polar Form.

  6. Derivative in Polar Form By writing the Polar equations in parametric form, we can derive the following: We use the product rule here. Don’t memorize this formula!

  7. Example Find the vertical tangent line to the following curve.

  8. At the pole, there may be NO tangent line, 2 or more tangent lines

  9. Area Inside a Polar Graph The length of an arc (in a circle) is given by r.q when q is given in radians. Imagine the area of our polar graph being broken up into infinitely small sectors. Then for a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:

  10. Example Find the area enclosed by:

  11. Example Find the area of the region inside the polar curve Area of one leaf times 4: Area of four leaves:

  12. Area between Two Polar Graphs To find the area between curves, subtract just like finding the areas between Cartesian curves, and establish limits of integration where the curves cross. Example: Find the area of the region inside the circle r = 1, but outside the curve r = 1 – cosθ.

  13. Arc length in Polar Form For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

  14. Example Use graphing calculator to find the arc length of one petal of the rose curve given by

  15. Surface Area in Polar Form When rotated about the x-axis: When rotated about the y-axis:

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