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Change of Variables II Reading Assignment: Read 5.5 again. Twice. Suggested problems: 5.5: 18, 19, 20,21, 23, 27,28,29, 31, 32 (hard!); Also: 5.8: 1, 3, 5, 9, 11, 12, 15, 17, 20 1) Find a transformation that will turn the triangle with vertices (0,0), (1,0), and (0,1) in the uvplane into the triangle with vertices (1,0), (0,1) and (2,2) in the xy-plane. 2) Find a transformation that will turn the unit square [0,1] [0,1] in the uv-plane into the triangle 0 y x 1 in the xy-plane. 3) Calculate 1 x 0 0 xy dy dx both directly and by changing variables as suggested in problem 2). 4) Calculate 5) Convert integrate. 2 4 x2 0 4 x2 1 /4 2 0 0 0 sin( x 2 y 2 ) dy dx . 3 sin(2 ) d d d into an integral in rectangular coordinates. Do not 6) Calculate 1 1 x 0 0 e y /( x y ) dy dx by making a helpful change of variables. (Ignore any difficulty this integral may have at the origin.) Hint: it would be nice to simplify the exponent, so why not take it all as one chunk? The x + y alone is causing troubles, so maybe it ought to be chunked together too. 7) Let R be the region in the first quadrant bounded by the hyperbolas xy = 1, xy = 3, x2 – y2 = 1, and x2 – y2 = 4. By choosing a convenient change of variables, calculate ( x2 y 2 ) dA R 8) Evaluate each integral. a. b. 9- x2 2 4 x2 3 0 0 0 0 2 0 x 2 y 2 dz dy dx 4 x2 y 2 0 z 4 x2 y 2 dz dy dx 9) Let D be the interior of the parallelogram with vertices at (0, 0), (2, 1), (3, 2), and (1,3) . Use dy dx a change of variables to simplify your task of evaluating . 1 x 2 y D (1,3) (3,2) D (0,0) (2,-1)