So I have the following integral:
$$I_1 = \iint u(x,y)dR$$
where $u(x,y)= e^{-(x^2 + y^2)}$
and the region $R$ is the rectangle $[-M,M]\times[-M,M]$.
I need to prove that $I_1$ equals:
$$I_2 = \left(∫_{-M}^Mu\left(\frac l{\sqrt2},\frac l{\sqrt2}\right)dl\right)^2.$$
I have been thinking for a long time and surfing the internet for integral properties that might help without any luck. I am really confused because I don't see how an area times an area will give a volume.
The only thing I'm positive about using is the fact that the integration variable is mute.
Where do I start? What can I try? or What hints can you give me?
$$\int_{-M}^{M} \int_{-M}^{M} e^{{-(x^{2}+y^{2}})} dxdy\\ =(\int_{-M}^{M} e^{-x^{2}}dx )(\int_{-M}^{M} e^{-y^{2}}dy)\\ =((\int_{-M}^{M} e^{-x^{2}}dx )^{2}$$ and $$\int_{-M}^{M} u(\frac l {\sqrt 2} ,\frac l {\sqrt 2})dl=\int_{-M}^{M} e^{-l^{2}}dl$$