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Using cylindrical coordinates to find center of mass of solid of uniform density given by $$x^2 + y^2 \le \frac{1}{4},\quad x^2 + y^2 + (z-1)^2 \le 1,\quad z \le 1.$$ I don't know how to set up the integrals. Can someone please help?

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Note that your solid is the intersection of a cylynder and a hemisphere. It is given by $$S:=\left\{(x,y,z)\in\mathbb{R}^3\::\; 1-\sqrt{1-(x^2+y^2)}\leq z\leq 1,\; x^2 + y^2 \leq \frac{1}{4}\right\}.$$ Moreover, by symmetry, the center of mass is along the $z$-axis.

Hence you need to evaluate the $z$-coordinate only: $$\bar z=\frac{\iiint_S z dxdydz}{\iiint_S dxdydz}$$ Use the cylindrical coordinates.

Can you take it from here?

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