So I've done some problems where a sphere intersects with a cylinder and I needed to find the volume of the intersected region using triple integrals. For example, if I needed to find the domain of integration for the intersection of the sphere $$x^2+y^2+z^2=a^2$$ and the cylinder $$x^2+y^2=\frac{a^2}{4}$$ then it would be $$D=\{(\rho, \phi, \theta)|0\le \rho \le a, 0 \le \phi \le \frac{\pi}{6}, 0 \le \theta \le 2\pi\}$$(adsbygoogle = window.adsbygoogle || []).push({});

Now let's say that the cylinders equation was something like $$(x-\frac{a}{2})^2+y^2=\frac{a}{2}$$ Now the cylinder is shifted by $$\frac{a}{2}$$, so how do I find the ranges for $$\phi$$ and $$\theta$$?

Side note: How do I do inline LaTeX?

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# Finding the domain of integration in spherical coordinate of a shifted cylinder

Can you offer guidance or do you also need help?

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