Find volume of $A=\{(x,y,z): x^2+y^2+z^2<1,(x-1/2)^2+y^2<1/4\}$

Question

$A=\{(x,y,z): x^2+y^2+z^2<1,(x-1/2)^2+y^2<1/4\}$

Find $$\iiint_A1dxdydz$$

I cannot think out good substitution to make Fubini work.

Answer 1

We have to find the volume of the intersection between a ball and a solid cylinder.
For such a purpose, it is enough to fix some $z\in[-1,1]$ and find the area $A_z$ of the intersection between the circle centered at the origin with radius $\color{purple}{\sqrt{1-z^2}}$ and the circle centered at $\left(\frac{1}{2},0\right)$ with radius $\color{green}{\frac{1}{2}}$.

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The angle between the green segments is $2\arcsin\sqrt{1-z^2}$, hence the angle between a green segment and the purple segment is $\arccos\sqrt{1-z^2}$. The area of the triangle delimited by the green and purple segments is $$ \frac{1}{8}\sin\left(2\arcsin\sqrt{1-z^2}\right) = \frac{|z|}{4}\sqrt{1-z^2}$$ hence: $$ A_z = \frac{1}{2}\arcsin\sqrt{1-z^2}+(1-z^2)\arccos\sqrt{1-z^2}-\frac{|z|}{2}\sqrt{1-z^2} $$ and the wanted volume is

$$ V = \int_{-1}^{1}A_z\,dz = \color{red}{\large\frac{2}{9}(3\pi-4)}.$$