Simplifying Double Integrals to Single-Variable Integrals Let D be a subset of $\mathbb{R}^2$ defined by $ |x| + |y| \leq 1$, and let $f$ be a continuous single-variable function on the interval $[-1,1]$. Show that
$$
\iint\limits_D \,f(x+y) \,  \mathrm{d}x \, \mathrm{d}y = \int_{-1}^{-1} \, f(u) \,  \mathrm{d}u
$$
This makes sense when you consider the region D since the values of x and y essentially range from -1 to 1 but I can't figure out a first solid step into the proof. Intuitively it looks plausible to me but that's it. Any help?

Write a change of variables, $u = x + y$, $v = x - y$. Then the Jacobian $J = \partial(x,y)/\partial(u,v) = 1/2$ and hence
$$\iint_D f(x+y) \ dx \ dy = \iint_D f(u) \ J \ du \ dv = \frac{1}{2} \int_{-1}^1\int_{-1}^1 f(u) \ du \ dv$$
Can you take it from here?
