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Thread: 3 questions about iterated integral

  1. #1
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    3 questions about iterated integral

    1) Suppose that $\displaystyle f_k$ is integrable on $\displaystyle [a_k,\;b_k]$ for $\displaystyle k=1,...,n$ and set $\displaystyle R=[a_1,\;b_1]\times...\times[a_n,\;b_n] $. Prove that $\displaystyle \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:


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


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

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

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


    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; Apr 14th 2013 at 12:08 AM.
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