3 questions about iterated integral

[ianchenmu]

Homework Statement

1) Suppose that $f_k$ is integrable on $[a_k,\;b_k]$ for $k=1,...,n$ and set $R=[a_1,\;b_1]\times...\times[a_n,\;b_n]$. Prove that $\int_{R}f_1(x_1)...f_n(x_n)d(x_1,...x_n)=(\int_{a_1}^{b_1}f_1(x_1)dx_1)...(\int_{a_n}^{b_n}f_n(x_n)dx_n)$

2)Compute the value of the improper integral:

$I=\int_{\mathbb{R}}e^{-x^2}dx$.

How to compute $I \times I$ and use Fubini and the change of variables formula?

3) Let $E$ be a nonempty Jordan region in $\mathbb{R}^2$ and $f:E \rightarrow [0,\infty)$ be integrable on $E$. Prove that the volume of $\Omega =\left \{ (x,y,z): (x,y) \in E,\;0\leq z\leq f(x,y)) \right \}$ satisfies

$Vol(\Omega)=\iint_{E}f\;dA$.

Homework Equations

n/a

The Attempt at a Solution

For (2), how to compute $I \times I$ and use Fubini and the change of variables formula? Perhaps (2) is the easiest to start with... but I have little idea for (1) and (3)...So thank you for your help.

 

[mfb]

(1) will rely on some other theorems, I think.

(2) is a standard method to calculate that improper integral. I x I gives
$$\iint e^{-x^2-y^2} dx dy$$, and with a change of the coordinate system this is easy to integrate.