Calculate the improper double integral: double integral
Calculate the improper double integral: $$ \iint_{\mathbb R^2} e^{-|x|-|y|}\,dx\,dy. $$
Explanation of Method
The integrand factors as a product of a function of $x$ alone and a function of $y$ alone: $e^{-|x|-|y|}=e^{-|x|}\cdot e^{-|y|}$. For such separable integrands, the double integral over $\mathbb R^2$ (or any product region) factors into a product of two single-variable integrals.
- Factor the integrand and the double integral into a product of single integrals.
- Evaluate $\displaystyle\int_{-\infty}^{\infty}e^{-|x|}\,dx$.
- Square the result.
Step 1 · Factor the double integral
$$ \iint_{\mathbb R^2} e^{-|x|-|y|}\,dx\,dy = \left(\int_{-\infty}^{\infty}e^{-|x|}\,dx\right)\left(\int_{-\infty}^{\infty}e^{-|y|}\,dy\right). $$
Step 2 · Evaluate the single integral
By symmetry, and splitting at $0$:
$$ \int_{-\infty}^{\infty}e^{-|x|}\,dx = 2\int_0^\infty e^{-x}\,dx = 2\Big[-e^{-x}\Big]_0^\infty = 2(0-(-1)) = 2. $$
Step 3 · Combine
$$ \iint_{\mathbb R^2} e^{-|x|-|y|}\,dx\,dy = 2\times2 = 4. $$
Additional Notes
- This factorization trick works because both the integrand ($e^{-|x|-|y|}=e^{-|x|}e^{-|y|}$) and the region ($\mathbb R^2=\mathbb R\times\mathbb R$) are Cartesian products — whenever this happens, a double integral collapses to a product of two ordinary single-variable integrals, avoiding polar coordinates or any other machinery entirely.
Result
The solution to the integral is: