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14 Notes 11

# 14 Notes 11 - MATH 20C Lecture 27 Monday Polar coordinates...

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MATH 20C Lecture 27 - Monday, November 29, 2010 Polar coordinates Recall: in the plane, x = r cos θ,y = r sin θ where r is the distance from the origin to the ( x,y ) point, θ is the angle with the positive x -axis. Drawn picture. Useful if either integrand or region have a simpler expression in polar coordinates. Area element: Δ A ( r Δ θ r (picture drawn of a small element with sides Δ r and r Δ θ ). Taking Deltar, Δ θ 0 , we get dA = r dr dθ. Example (from way back in Lecture 22): ZZ x 2 + y 2 1 , 0 x 1 , 0 y 1 ( 1 - x 2 - y 2 ) dxdy = Z π/ 2 0 Z 1 0 (1 - r 2 ) r dr dθ = Z π/ 2 0 ± r 2 2 - r 4 4 ² r =1 r =0 = π 8 . Once again, ZZ R f ( x,y ) dA = ZZ R f ( r,θ ) r dr dθ. In general: when setting up RR fr dr dθ, ﬁnd bounds as usual: given a ﬁxed θ, ﬁnd initial and ﬁnal values of r (sweep region by rays). Cylindrical coordinates ( r,θ,z ) where x = r cos θ,y = r sin θ. (Drawn picture.) Here r measures distance from z -axis, θ measures angle from xz -plane, z is still the height. Cylinder of radius 8 centered on

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14 Notes 11 - MATH 20C Lecture 27 Monday Polar coordinates...

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