2224-Sec13_4-HWT

2224-Sec13_4-HWT - Math 2224 Multivariable Calculus Sec....

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Unformatted text preview: Math 2224 Multivariable Calculus Sec. 13.4: Double Integrals in Polar Form I. Introduction A. Formula Review x = r cos ! y = r sin ! x 2 + y 2 = r 2 B. Integrals in Polar Coordinates In polar coordinates, the natural shape to divide R into is a polar rectangle whose sides have constant r- and -values. ! A k = area of large sector - area of small sector ! A k = 1 2 r k + ! r 2 " # $ % & 2 ! ( ) 1 2 r k ) ! r 2 " # $ % & 2 ! ( = ! ( 2 r k + ! r 2 " # $ % & 2 ) r k ) ! r 2 " # $ % & 2 * + , ,- . / / ! A k = ! ( 2 r k 2 + r k ! r + ! r 2 ( ) ) r k 2 ) r k ! r + ! r 2 ( ) * +- . = ! ( 2 r k 2 + r k ! r + ! r 2 ) r k 2 + r k ! r ) ! r 2 * +- . ! A k = ! ( 2 2 r k ! r [ ] = r k ! r ! ( ! " A = r k " r " # =area of the k th piece, where r k is the midpoint ! dA = rdrd " f ( r , ) dA R = f ( r , ) rdrd R C. Area in Polar Coordinates The area of a closed and bounded region R in the polar coordinate plane is rdrd R ....
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2224-Sec13_4-HWT - Math 2224 Multivariable Calculus Sec....

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