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Unformatted text preview: Compute double integrals in polar coordinates Useful facts : Suppose that f ( x, y ) is continuous on a region R in the plane z = 0. (1) If the region R is bounded by and a r b , then Z Z R f ( x, y ) dA = Z beta Z b a f ( r cos , r sin ) rdrd. (2) If the region R is bounded by and r 1 ( ) r r 2 ( ) (called a radially simple region), then Z Z R f ( x, y ) dA = Z beta Z r 2 ( ) r 1 ( ) f ( r cos , r sin ) rdrd. Example (1) Find the volume of a sphere of radius a by double integration. Solution: We can view that the center of the sphere is at the origin (0 , , 0), and so the equation of the sphere is x 2 + y 2 + z 2 = a 2 . We then can compute the volume of the upper half part of the sphere and multiply our answer by 2. V = 2 Z a- a Z a 2- y 2- a 2- y 2 q a 2- x 2- y 2 dxdy. To compute this integral, we observe that the polar coordinates may be a better mechanism in this case. With polar coordinates, the functionin this case....
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