Question
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.
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.
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.
Answers
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
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
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