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MTH 253 Calculus (Other Topics) Chapter 10 – Conic Sections and Polar Coordinates Section 10.7 – Areas and Length in Polar Coordinates Copyright © 2009 by Ron Wallace, all rights reserved.

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Integrals – A reminder … A definite integral is a limit of a Riemann sum.

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Areas in Polar Coordinates Fan shaped regions bounded by a polar curve and two rays emanating from the pole (i.e. origin).

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Areas in Polar Coordinates 1.Divide the region into n pieces with evenly spaced rays emitting from the pole. 2.Approximate the area of each piece using a sector of a circle. 3.Add up all of the areas of the sectors. 4.Take the limit as the number of sectors approaches infinity. 5.Which gives an integral.

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Areas in Polar Coordinates Example: Find the area inside one pedal of the curve

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Areas Between Polar Curves Subtract Regions

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Areas Between Polar Curves Subtract Regions Yellow Area:

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Areas Between Polar Curves Subtract Regions Yellow Area: Pink Area:

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Areas Between Polar Curves Subtract Regions Yellow Area: Pink Area: Green Area (between the curves) = Yellow Area – Pink Area NOTE: Use either form, whichever make sense to you!

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Areas Between Polar Curves Example: Find the area between the curves … Red Area: Total Area:

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Lengths of Polar Curves Curves bounded two rays emanating from the pole (i.e. origin).

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Length of a Curve - Review yy xx s (a,b) (c,d) 1 of 2

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Length of a Curve - Review 2 of 2 ab m & n are either values of x, values of y, or values of t depending on how the function is defined (explicitly or parametrically).

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Lengths of Polar Curves

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Example: Find the length of the spiral …

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