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Section 10.5 Area and Arc Length in Polar Coordinates

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Area Enclosed by Polar Curves Similar to Cartesian equations, we can find the exact area of the polar region using an integral. The only exception is that we are using sectors to approximate the area, not rectangles. The area of an sector is: The area the highlighted sector is: If we integrate this area over the entire interval, it represents the total area bounded by the curve.

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Area Enclosed by Polar Curves

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Example 1 The area is therefore:

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Example 2 The area is therefore:

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Example 3 If you created a graph, made a table, or analyzed the equation; you will see the size of each loop are identical. Instead of finding the whole area, you could triple the area of one loop. Find where the curve goes through the pole: Use consecutive values for the interval and triple the integral:

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Area Enclosed by Polar Curves It is still outside curve minus the inside curve.

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Example If you created a graph, made a table, or analyzed the equations; you will see they intersect twice. Find that intersection:

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Warning! NEVER ASSUME ANYTHING ABOUT A POLAR CURVE

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Example If you created a graph, made a table, or analyzed the equations; you will see they intersect once. Find that intersection:

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Example (Continued)

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Rule of Thumb for Polar Curves Never ASSUME anything about the curve. Always check the graph. Ask yourself Do the intersections have the same angle? How many cycles are being graphed? Am I seeing the complete picture? Does my change in theta need to be decreased? Etc.