l_notes16 - Lecture XVI Integrals in Polar Cylindrical or...

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± ± Lecture XVI Integrals in Polar, Cylindrical, or Spherical Coordinates Usually, we write functions in the Cartesian coordinate system. Hence we write and compute multiple integrals in Cartesian coordinates. But there are other coordinate systems that can help us compute iterated integrals faster. We analyze bellow three such coordinate systems. 1 Polar coordinates In E 2 , the polar coordinates system is often used along with a Cartesian system. The polar coordinates of a point P ( x, y ) are r and θ where r is the distance from the origin O and θ is the angle done by OP and Ox measured counter-clockwise from Ox . The equations we use to switch from one system to the other are: x = r cos θ, y = r sin θ, r = x 2 + y 2 . To transform an integral R f ( x, y ) dA from Cartesian coordinates to polar coordinates, we take a look at Reimann sums. By consider a particular type of subregions, we get that dA = rdrdθ , so we can write: ² ² ² ² f ( x, y ) dA = f ( r cos θ, r sin θ ) rdrdθ, ˆ R R where
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This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.

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l_notes16 - Lecture XVI Integrals in Polar Cylindrical or...

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