An alternative: from the parametric equations of the cylinder,
and substituting in the equation of the sphere we obtain
I'm looking to parametrize the upper surface of a portion of the sphere x^2 + y^2 + z^2 = 16 contained within the cylinder x^2+y^2 = 4y
since its a sphere, i've parametrize into spherical coordinates with
x(u,v) = (4sin(u)cos(v), 4sin(u)cos(v),4cos(u))
however, I'm stuck on finding the bounds for u. specifically with the cylinder being x^2+(y+1)^2 = 1, i'm unsure how to set u. i have r = 4sinv, but that doesn't seem to help. Maybe i'm going about the question wrong.
how would i solve the integral.
I've simplified the integral to 4(int(1/(16-x^2-y^2)) dxdy
with a D of x^2+y^2-4y I can do a change of variable and end up with
2u = x and 2v-2 but i'll end up with a very messy integral but ranges of [0,2pi]x[0,1]
or i can substitute x=rcos*u and y = rsin*u and end up with ranges of [0,2pi]x[0,4sin*u] and an integrand of 1/(16-r^2). and if i continue on, i'll end up arcsin(r/4) and thus a complicated integral.
is there a way i can paramatize it into spherical coordinates and find ranges for phi? because that would probably make this question more doable...