Calculus Notes 11.4

Calculus Notes 11.4 - Calculus-Stewart Dr Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 11.4 11.4 Areas and Lengths in Polar Coordinates Area To develop a means of finding the area using polar coordinates, we begin by observing that the area of a sector of a circle of fixed radius r is proportional to its central angle. Thus A = θ 2 π r 2 ( ) = 1 2 r 2 . Theorem The area contained in the polar region bound by the rays a and b and the curve r = f ( ) is given by A = 1 2 r 2 d a b = 1 2 f ( ) [ ] 2 d a b . Proof: Partition the angle between a and b into small sectors A i . If the sector is small enough the radius is almost constant and the area is approximated by A i 1 2 f ( i * ) [ ] 2 d where i * is any angle in the i th sector and d is the angle for that sector. Thus, the total area is approximated by A A i i = 1 n = 1 2 f ( i * ) [ ] 2 d i = 1 n . Taking the limit as n goes to infinity gives the desired result. Example A Find the area enclosed by one loop of the four-leaved rose r = cos2 .
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Calculus- Stewart Dr. Berg Summer ‘09 Page 2 11.4
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This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas.

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Calculus Notes 11.4 - Calculus-Stewart Dr Berg Summer 09...

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