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**1. The problem statement, all variables and given/known data**

Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y##

**2. Relevant equations**

**3. The attempt at a solution**

I have already posted this question on MSE: https://math.stackexchange.com/questions/3155620/surface-integral-of-an-intersection-cone-plane/3155634?noredirect=1#comment6498746_3155634

My issue is with ##\iint_S ydS =\sqrt{3} \int_A ydxdy=\sqrt{3}\, \bar{y}|A|##.

**Concretely, I do not get why ##\bar{y}## shows up.**

My issue is that I still do not understand how to deal with the argument of the surface integral. Let's say we had for instance ##\iint_S xydS## or ##\iint_S y^2dS##. I wouldn't know how to proceed. I know it is somehow related to the centroid of the figure (in this case an elliptical cylinder).

Robert Z provided a short explanation but I do not get it...

May you please provide an explanation?

Thanks