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10th - d should show what the variables are in the...

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INTEGRATION PROBLEMS, MATH  , WEEK  DAVID PIERCE . Find R f when R = [0 , 2] × [1 , 2] and f ( x, y ) = 3 xy 2 - 2 x 2 y . . Find the average value of y 2 / ( x 2 + 1) when 0 x 1 and 0 y 1 . . Express as an iterated integral the volume enclosed by: the two sheets of the hyperboloid given by 4 x 2 - 4 y 2 - z 2 = 4 ; the XY - and XZ -planes; the planes given by z = 2 and y = 3 . . Find R f when R = { ( x, y ): 0 y π & 0 x 1 + cos y } and f ( x, y ) = x 2 sin y . . Write R f as an iterated integral in two ways when R is the quadrilateral with vertices (0 , 0) , (4 , 0) , (3 , 2) , and (0 , 3) . . Reverse the order of integration in e 1 ln y 0 f ( x, y ) d x d y . . Compute 2 0 4 x 2 12 x 3 exp( y 3 ) d y d x . . Using symmetry, find the volume enclosed by: the hyperbolic paraboloid x = y 2 - z 2 ; the plane x = 1 ; the cylinder y 2 + z 2 = 1 . Remark. Where Stewart writes R f ( x, y ) d A , I write R f here, on the basis that: (a) considered as integrals, both areas and volumes are the same thing: they are limits of sums of sizes of rectangular figures; (b) the number of signs d (if used at all) should match the number of signs ; (c) what comes after the sign
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Unformatted text preview: d should show what the variables are in the integrand; (d) the notation R R f or R R f ( x,y ) d( x,y ) will be used in a rigorous treatment of the mathematics, as in [  ,  ]. References [  ] Tom M. Apostol. Mathematical analysis . Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second edition,  . [  ] Michael Spivak. Calculus on manifolds. A modern approach to classical theorems of advanced calculus . Addison-Wesley Publishing Company, Reading, Massachusetts,  . Corrected reprint of the  edition. Mathematics Department, Middle East Technical University, Ankara  , Turkey E-mail address : [email protected] URL : http://www.math.metu.edu.tr/~dpierce/ Date : April  ,  ....
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