- #1

vDrag0n

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## 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.