Unformatted text preview: RR D xydA, where D : x ≥ 1 ,y ≥ , ( x1) 2 + y 2 ≤ 1 . Example 3 R ∞∞ ex 2 dx. Denote by A our integral. It will be nonnegative since the exponential is positive. Then A 2 = A · A = ±Z ∞∞ ex 2 dx ²±Z ∞∞ ey 2 dy ² = Z ∞∞ Z ∞∞ ex 2y 2 dxdy. Changing to polar coordinates, this gives A 2 = Z 2 π Z ∞ rer 2 dr. The inner integral is equal, via the change of variables u = r 2 , to 1 2 Z ∞ eu du = 1 2 . Hence A 2 = π, and A = √ π. Example 4 Let W : x 2 + y 2 ≤ 1 , ≤ z ≤ 1 + x 2 + y 2 . ZZZ W ( x 2 + y 2z ) dV = Z 2 π Z 1 Z 1+ r 2 ( r 2z ) rdzdrdθ = ... Example 5 The volume of the unit ball B in R 3 can be computed using cylindrical coordinates (4 π/ 3). 1...
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This note was uploaded on 04/28/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
 Spring '08
 Helton
 Math

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