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Math Help - 3 questions about iterated integral

  1. #1
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    3 questions about iterated integral

    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)d  x_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.
    Last edited by ianchenmu; April 14th 2013 at 01:08 AM.
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