Question

Use a double integral to find the volume. R=[0,2] x [0,2]. f(x,y) = e^ ( -(x+y)/4 )

Answer 1

\[\begin{align*}\iint_Rf(x,y)\,dx\,dy&=\int_0^2\int_0^2\exp\left(-\frac{x+y}{4}\right)\,dx\,dy\\\\
&=\int_0^2\int_0^2\exp\left(-\frac{x}{4}\right)\exp\left(-\frac{y}{4}\right)\,dx\,dy\\\\
&=\left(\int_0^2\exp\left(-\frac{x}{4}\right)\,dx\right)\left(\int_0^2\exp\left(-\frac{y}{4}\right)\,dy\right)\\\\
&=\left(\int_0^2\exp\left(-\frac{x}{4}\right)\,dx\right)^2\end{align*}\]

Answer 2

Argh, I never fail to forget that property. Thanks for clearing that up