Solution Found!
Cylinder and sphere Consider the sphere x2 + y2 + z2 = 4
Chapter 11, Problem 55E(choose chapter or problem)
Consider the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \((x-1)^{2}+y^{2}=1\), for \(z \geq 0\).
a. Find the surface area of the cylinder inside the sphere.
b. Find the surface area of the sphere inside the cylinder.
Questions & Answers
QUESTION:
Consider the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \((x-1)^{2}+y^{2}=1\), for \(z \geq 0\).
a. Find the surface area of the cylinder inside the sphere.
b. Find the surface area of the sphere inside the cylinder.
ANSWER:Solution 55EStep 1 The equation of given sphere is and of cylinder is (a)It is required to find the surface area of cylinder inside the sphere. Consider the equation of cylinder which is an infinite cylinder . Since, the radius of sphere is 2 and the radius of base of cylinder is 1 with its base center at (1, 0), base of cylinder completely lies inside of the sphere and touches surface of the sphere. Therefore, it is required to find area of partial cylinder.Area of the required cylinder below sphere and above can be obtained using parametric description of cylinder. The equation of cylinder in cylindrical coordinates is for .So, parameterizing cylinder gives The challenge is in finding the limits on which depends on . The sphere in cylindrical coordinates is given as Putting to get relation between Therefore, region of integration is Therefore, required surface area is The area calculated above is half the area of required cylinder as it is projection on one plane. Summing for two projections along z-y plane and z-x plane, the required area of cylinder below sphere and above is 8.