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Unformatted text preview: (1 , , − 2) and (4 , 6 , 3) . 4 32B MIDTERM 2 RADKO Problem 3. Rewrite the integral i i i E f ( x, y, z ) dV , where E = { ( x, y, z )  x 2 + y 2 + z 2 ≤ a 2 ; x 2 + y 2 ≤ z 2 } in spherical coordinates. (Here f = f ( x, y, z ) is a given function and a > is a given number). 32B MIDTERM 2 Radko 5 Problem 4. Use the transformation x = √ 2 u − r 2 / 3 v , y = √ 2 u + r 2 / 3 v to evaluate the integral i i R ( x 2 − xy + y 2 ) dA , where R is the region bounded by the ellipse x 2 − xy + y 2 = 2 . 6 32B MIDTERM 2 RADKO Problem 5. Let C be a positivelyoriented smooth simple closed curve on the xyplane such that the area enclosed by this curve is equal to A . Use the Green’s formula to show that the area enclosed by the curve can be computed as A = 1 2 i C xdy − ydx ....
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 Fall '08
 Rogawski
 Math

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