Why do we only need a double integral to compute the centre of mass in this problem

Question

I am confused with the following;

Asked to find the centre of mass of the part of the sphere $x^2+y^2+z^2=25$ above the plane $z=4$.

In the solutions, it uses that $$M=\iint_{S} \delta dS$$ where \delta is constant.

But I am confused, I thought when calculating centre of mass in three dimension we use the triple integral over the volume

Can someone please help to explain this to me why they are equivalent?

Answer 1

You should use a triple integral, indeed, when the solid is 3-dimensional.

But here, you are dealing with a surface, which is a 2-dimensional solid (in a 3-dimensional space).

Perhaps you misunderstood the difference between a sphere and a ball.