Hey I have problems with this problem.
Let $\mathbb{R}^3$ be a sphere, described by $x^2 +y^2 +z^2 ≤ R^2$ , and a cylinder, described by $(x − R/2 )^2 + y^2 ≤ ( R/2 )^2$ .
a) Calculate the part of the sphere that lies inside the cylinder.
b) Calculate the part of the cylinder surface that lies inside the sphere.
To calculate the volume of the part of the sphere that lies inside the cylinder, we can evaluate the double integral of the sphere over the region defined by the cylinder:
$V = \iint_{(x-R/2)^2 + y^2 \leq (R/2)^2} \sqrt{x^2 + y^2 + z^2} dxdy$
Similarly, to calculate the area of the part of the cylinder surface that lies inside the sphere, we can evaluate the surface integral of the cylinder over the region defined by the sphere:
$A = \iint_{x^2 + y^2 + z^2 \leq R^2} \sqrt{(x-R/2)^2 + y^2} dx dy$
But from here I don't know how to proceed. Can someone help me?
In a more general way? How should I proceed when I have such integrals (volume of something under some conditions)
Outgoing from comment above integral will be $$4\int\limits_{0}^{R}\int\limits_{0}^{ \sqrt{( R/2 )^2-(x − R/2 )^2} }\int\limits_{0}^{\sqrt{R^2-x^2-y^2}}$$ So, bounds you have. Can you finish?