We wish to compute the double integral cos((x-y)/(x+y))dA where R is the first quadrant triangle beneath the line x+y=1.
a) Let s=x+y and t=x-y. Sketch the region R with labeled level curves of s and t sketched in. Answer the following questions to set up limits of integration in the dtds order:
(i) When x=0, what is t in terms of s? When y=0, what is t in terms of s?
(ii) What is the value of s at the lower left-hand corner of the region? What is the value of s on the side determined by x+y=1? Now set up the limits of integration and the integrand.
b) Solve for x and y in terms of s and t and compute the absolute value of the Jacobian in order to obtain the st version of dA.
c) Compute the original integral as an st integral.