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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?

rmzep
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1 Answers1

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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?

Simon S
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