Let \(D'\) be the region in the \(xy\)-plane bounded by the lines \(y=0\text{,}\)\(x=0\text{,}\) and \(x+y=1\text{.}\) We will evaluate the double integral
We would like to make a substitution that makes the integrand easier to antidifferentiate. Let \(s = x+y\) and \(t = x-y\text{.}\) Explain why this should make antidifferentiation easier by making the corresponding substitutions and writing the new integrand in terms of \(s\) and \(t\text{.}\)
Solve the equations \(s = x+y\) and \(t = x-y\) for \(x\) and \(y\text{.}\) (Doing so determines the standard form of the transformation, since we will have \(x\) as a function of \(s\) and \(t\text{,}\) and \(y\) as a function of \(s\) and \(t\text{.}\))
Make the change of variables indicated by \(s = x+y\) and \(t = x-y\) in the double integral (12.8.4) and set up an iterated integral in \(st\) variables whose value is the original given double integral. Finally, evaluate the iterated integral.
