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Consider the double integral Sex 1(a,x)dA where f(x, y) = sin(x - y) and R is the parallelogram shown in the figure below. 0 1 2 3 -1 - 2 -3 The equations of the lines for the boundary edges of the parallelogram are: y = 2 y=2-5 3 y = 2 NICONICO Y= 2+ 2 However, instead of solving the integral directly (which would involve splitting it into three sections) you are to use the transformation U=I-Y 2 - y - 2
a(x,y) (a) First, calculate the absolute value of the determinant of the Jacobian matrix a(u, v) You must enter your answer as a fraction, not a decimal, as appropriate. (b) What is the the function F(u, v)? (c) The new limits of integration are determined via the fact that boundary curves of R in the (2,y)-plane are transformed into the boundary curves for R* in the (u, v)-plane. For the integral in the form D Si Se F(1,0) || dvdu, enter the limits of integration below. You must enter your answer as a fraction, not a decimal, as appropriate. A= B= C= 6 D= (d) Hence calculate the value of the double integral. You must give your answer to exactly 4 significant figures. Number