$$\iint_Sf(x,y,z)\,dS\,,$$ where $S$ is the part of graph $z=x^2+y^2$ below the plane $z=y$.
I am wondering what is the surface mean. I can not imagine it. If I use the polar coordinates, then what is the range of each variables?
2026-03-28 10:41:36.1774694496
How to investigate the surface integral $\iint_Sf(x,y,z)\,dS$?
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Intersection of $z=x^2+y^2$ and $z=y$ is $\left(y- \frac{1}{2} \right)^2+ x^2 = \frac{1}{4}$. $$\underset{S}\iiint f(x,y,z)dS = \underset{\left(y- \frac{1}{2} \right)^2+ x^2 \leqslant \frac{1}{4}}\iint f(x,y,x^2+y^2)\sqrt{1+4x^2+4y^2}dxdy$$