Parametrize and find the surface area

[vDrag0n]

Homework Statement

a)parametrize the upper surface of that portion of the sphere x^2 + y^2 +z^2 = 4 contained within the cylinder, x^2 + y^2 = 2y
b)find the area of the surface

Homework Equations

how to find the range for phi. for paramaetrizing into spherical coordinates

The Attempt at a Solution

a) because its a sphere i tried parmetrizing into spherical coordinates
and thus
X (u,v) = (2sinucosv, 2sinusinv,2cosu)
u,v = [0,?]x[0,2pi]
I've tried substuting the eq of the cylinder into the eq of the sphere but i ended up with
z^2 = 4-2y => (4 - 4sinusinv)^0.5
which isn't very helpful in finding the ranges of U



b) without finding the ranges for u, I won't be able to use the formula doubleint ||Xu x Xv|| du dv

but i can try A(S) = double int (1 + ||gradF||^2)^0.5 dx dy
i can have F = (4-x^2 - y^2)^0.5 and D == x^2 + y^2 - 2y <=0
looks like polar coordinates would be best so i set
x = r cos u
y = r sin u
r,u = [0,2sinu] x [0, 2pi]
and thus end up with an integral of
int ( 2/ (4 - r^2)^0.5 dr du
but this will give me an integral of
int 2 *( [arcsin (r/2)] from 0 to rsin ) du which is not an integral i would like to solve.

 

[vDrag0n]

I've rethought this and here's my try.

http://img717.imageshack.us/img717/6788/answer.png

Uploaded with ImageShack.us

 

[SammyS]

Homework Statement

a)parametrize the upper surface of that portion of the sphere x^2 + y^2 +z^2 = 4 contained within the cylinder, x^2 + y^2 = 2y
b)find the area of the surface

Hi vDrag0n.

In cylindrical coordinates, x² + y² = r² and y=r·sinθ .

So the equation for the cylinder is: r² = 2·r·sinθ → r = 2·sinθ.

The equation for the sphere is: r² +z² = 4, so the upper half is:

[tex]z=\sqrt{4-r^2}\,.[/tex]