<s>[INST] triple integral to calculate volume Let $V$ be the volume of the $3$-dimensional structure bounded by the paraboloid $z=1−x^2−y^2$, planes $x\ge0, y\ge0$ and $z\ge0$ and by the cylinder $x^2+y^2−x\le0$. If $V=a\pi/b$, where $a$ and $b$ are coprime positive integers, what is the value of $a+b$?
if we consider $x^2+y^2=r^2,x=r\cos x,y =r\sin x$, we get $0\le z\le 1-r^2, 0\le r\le\cos x$ and$0\le x\le\pi/2$.
Is anything wrong with the limits?
 [/INST] The cross-section of this volume is the intersection of the circles $(x-1/2)^2+y^2=1$ and $x^2+y^2=1-z$.  Draw the picture to see that the intersection breaks into two pieces: that bounded on top by the first circle, and that by the second.  The integral is
$$\int_0^1 dz \: \left [ \int_0^{1-z} dx \: \int_0^{\sqrt{1/4 - (x-1/2)^2}} dy + \int_{1-z}^{\sqrt{1-z}} dx \: \int_0^{\sqrt{1-z - x^2}} dy  \right ]$$
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