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Show that $\int^1_0 \int^{1-x}_0e^\frac{y}{x+y}dy\,dx=\frac{e-1}{2}$

This is probably very simple, but I was instructed to use the substitution $x+y=u, y=uv$.

So by manipulating the integral and calculating the Jacobian (which is simply $u$, ie. $dx\,dy=u\,du\,dv$, I can express the transformation as $$\int \int ue^v dv \,du$$

The problem is that I don't know what to plug in as the limits of integration. For example, the limit $1-x$ in the expression $\int^1_0 \int^{1-x}_0e^\frac{y}{x+y}dy\,dx$ cannot be expressed as either a function of v or u purely with the substitution given. Am I missing something here? Thanks for the help.

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3 Answers 3

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  1. the original domain is the right triangle formed by (0,0), (1,0), (0,1) on the x-y plane
  2. For $u=x+y$, consider $u$ as a new axis along the line x-y=0.For the triangle domain, $u \in [0,1]$
  3. From $u=x+y$ again, if you work on the contours of $x+y$ for a few values of u, they are all straight line with slope -1. If you change u in $[0,1]$, you are moving this contour line in the triangle domain. In particular, each such contour line has end points cutting the triangle edges at $v=0$ and $v =1$.
  4. So, $0\leq u\leq 1$ and $0\leq v \leq 1$ is the new domain you need.
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  • $\begingroup$ Opssss...I don't know why I've assumed it was a square! I fix that. $\endgroup$
    – user
    Sep 4, 2018 at 16:57
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Since on $x-y$ plane $x+y=k$ represents a line parallel to the line $y=-x$ we have that

  • $u=x+y \implies 0\le u\le 1$

then for any $u$ we have

  • $0\le y\le u\implies 0\le v\le 1$
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    $\begingroup$ You made a mistake, $|J|=u-uv+uv=u$. $\endgroup$
    – alans
    Sep 4, 2018 at 16:25
  • $\begingroup$ opsssss....I fix. Thanks $\endgroup$
    – user
    Sep 4, 2018 at 16:28
  • $\begingroup$ I'm trying to understand the steps. Can you please help me understand the steps: First I want to map the region in xy plane to uv plane? So I try to map $y$ axis ($x=0$) first: $x=0=u(1-v)\implies y-$ axis is mapped onto a pair of straight lines. But this would mean that the mapping $u=x+y, y=uv$ is not 1-1 so why are we allowed to use this substitution. Thanks. $\endgroup$
    – Koro
    Jan 18, 2022 at 14:21
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    $\begingroup$ @Koro We are mapping the triangular region in $xy$ plane to a rectangular region in $uv$ plane and this is $1-1$. You can check this pictorially. Refer also to the related OP here. $\endgroup$
    – user
    Jan 22, 2022 at 10:39
  • $\begingroup$ @user: Thanks. I found an alternative to the tranformation so I went ahead with that. :) $\endgroup$
    – Koro
    Jan 22, 2022 at 11:32
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Hint: Limits of integration are 0 and 1.

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