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.