3 questions about iterated integral

[i_a_n]

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

$Vol(\Omega)=\iint_{E}f\;dA$.Perhaps (2) is the easiest to start with... but I have little idea for (1) and (3)...So thank you for your help.

 

[Petrus]

Hello http://www.mathhelpboards.com/members/ianchenmu/,
1) I would use the rule:$$ f(x,y)=g(x)h(y)$$ where $$R = [a,b]$$ x $$[c,d]$$ hence:
$$\int\int_R g(x)h(y) dA= \int_a^b g(x)dx\int_c^d h(y) dy$$
cause if we think like this
$$\int\int_R f(x,y)dA = \int_c^d\int_a^b g(x)h(y) dxdy= \int_c^d\left[ \int_a^b g(x)h(y) dx \right]dy$$ notice we used Fubini's theorem, notice that $$h(y)$$ is a constant and then we can take it out!
I am not really good on explain but I hope you understand.

Regards,

 

[Ackbach]

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?

The standard way is as follows:
$$I^{2}= \int_{ \mathbb{R}}e^{-x^{2}} \,dx \cdot \int_{ \mathbb{R}}e^{-y^{2}} \,dy= \int_{ \mathbb{R}} e^{-(x^{2}+y^{2})} \, dx \, dy.$$
Then change to polar coordinates and see what happens.