# Thread: parametrize surface of a sphere contained within a cylinder

1. ## parametrize surface of a sphere contained within a cylinder

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))
with v={0,2pi}

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.

2. An alternative: from the parametric equations of the cylinder,

$C \equiv\begin{Bmatrix}x=2\cos t\\y=2+2\sin t\\z=\lambda\end{matrix} \quad (t\in[0,2\pi],\;\lambda\in\mathbb{R})$

and substituting in the equation of the sphere we obtain

$\lambda^2=8(1-\sin t)$

Fernando Revilla

3. thanks for you help. I forgot to mention that i'd like to find the area of the surface also.

4. Originally Posted by vicariage
thanks for you help. I forgot to mention that i'd like to find the area of the surface also.

Of the cylinder? . Of the sphere? . Limited by ...

Fernando Revilla

5. Originally Posted by FernandoRevilla
Of the cylinder? . Of the sphere? . Limited by ...

Fernando Revilla
of the upper portion of the sphere contained within the cylinder

6. Originally Posted by vicariage
of the upper portion of the sphere contained within the cylinder
Then,

$S=\displaystyle\iint_{D}\sqrt{1+\left(\frac{{\part ial z}}{{\partial x}}\right)^2+\left(\frac{{\partial z}}{{\partial y}}\right)^2}\;dxdy\;,\quad z=\sqrt{16-x^2-y^2}$

where:

$D \equiv x^2+y^2-4y \leq 0$

Fernando Revilla

7. 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...