Cube Sphere Intersection

I will rearrange things a bit to make it easier to "wrap my head around" this problem. Let's place the center of the sphere at $(0,0,0)$ and the center of the cube at $(0.5,1.5,1)$ (I'm permuting the coordinates). Now we seek the volume of the portion of the sphere with $1.5 \geq x \geq -0.5, y \geq 0.5$ and $z \geq 0$ , which is simply the double integral $\int_{-0.5}^{1.5}\int_{0.5}^{\sqrt{4-x^2}}\sqrt{4-x^2-y^2}\ dy\ dx \approx 3.551$ . With the volume of the sphere being $\frac{32\pi}{3}\approx 33.51$ , the required percentage comes out to be about $\boxed{10.6}\%$ .

This is a lovely problem indeed: Not hard at all once one understands what is going on. Thank you for sharing!