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3 questions about iterated integral

  1. Apr 14, 2013 #1
    1. The problem statement, all variables and given/known data


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

    2)Compute the value of the improper integral:

    [itex]I=\int_{\mathbb{R}}e^{-x^2}dx[/itex].

    How to compute [itex]I \times I[/itex] and use Fubini and the change of variables formula?

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

    [itex]Vol(\Omega)=\iint_{E}f\;dA[/itex].

    2. Relevant equations

    n/a

    3. The attempt at a solution

    For (2), how to compute [itex]I \times I[/itex] 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.
     
    Last edited: Apr 14, 2013
  2. jcsd
  3. Apr 14, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    (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.
     
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