Find the surface area of the part of the sphere x^2+y^2+z^2=a^2 inside the circular cylinder x^2+y^2=ay (r=a*sin(θ) in polar coordinates), with a>0.

Question

First time posting on this website, sorry for the lack of details on my attempts but I am really not sure where to start on this problem.

A formula that is useful is A(G)=∫∫√((f_x)^2+(f_y)^2+1)dA

*f_x is the partial derivative with respect to x, f_y is the partial derivative with respect to y

I know that I need to find an equation which should be x^2+y^2+z^2=a^2, and I need to find the limits which is where I am really struggling.

Also according to my professor, I shouldn't have to use any polar coordinate conversions in order to complete this problem, which was my initial approach.

Answer 1

The circle in the x-y plane is

x^2 + (y - a/2)^2 = a^2/4
x^2 = a^2/4 - (4y^2-4ay+a^2)/4 = (y^2-ay)/4

The limits of integration in the x-y plane are

0 < y < a
0 < x < (1/2)√(y^2-ay)

then use symmetry and multiply by 4