Find the integral $$\int\int _{[0,1] \times[0,1]} \max \left\{x,y\right\} dx \,dy$$
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This is not a surface integral, but simply an integral over a 2D region. To evaluate it, simply break the integral up into two pieces: one where $x \gt y$, and vice-versa. The integral is then equal to
$$\int_0^1 dx \, x \, \int_0^x dy + \int_0^1 dx \, \int_x^1 dy \, y = \frac13 + \frac12 \cdot \frac13 = \frac12$$
Ron Gordon
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@ Ron : How have you break this integeral into two pieces, Please cleared me. – Struggler Jan 03 '16 at 16:26
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@Struggler: by considering the max function. Plot it over the given region and you will see how the integral breaks up this way naturally. – Ron Gordon Jan 03 '16 at 16:28
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Thank you so much for the prompt reply – Struggler Jan 03 '16 at 16:35