Hi, I would like some help with the following question.(adsbygoogle = window.adsbygoogle || []).push({});

Q. Let f be continuous on [0,1] and let R be the triangular region with vertices (0,0), (1,0) and (0,1). Show that:

[tex]

\int\limits_{}^{} {\int\limits_R^{} {f\left( {x + y} \right)} dA = \int\limits_0^1 {uf\left( u \right)} } du

[/tex]

By making the substitutions u = x+y and v = y, I got it down to:

[tex]

\int\limits_{}^{} {\int\limits_R^{} {f\left( {x + y} \right)} dA = } \int\limits_0^1 {\int\limits_0^u {f\left( u \right)} dvdu}

[/tex]

The above leads to the given result. However, I was stuck on trying to get bounds for the integrals so I'm not sure if I've justified those limits of integration properly.

From the boundary line y = 1 - x, the substitution u = x+y gives u = 1. From the boundary line x = 0, the substitutions yield u = 0 + y = v so that v = u. The boundary line y = 0 yields v = 0. The lower limit for u is what I am lacking in. The three boundaries that I've just obtained completely describe the region R anyway so I decided to say that at the origin x,y = 0 which gives u = 0. I'm not sure whether I should've done something else to obtain the lower u limit.

Any help would be good.

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# Homework Help: Change of variables

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