Surface area of cap using integrals

[lilmul123]

Homework Statement

The question asks,

"Find the surface area of the cap cut from the sphere x^2+y^2+z^2=2 by the cone z = sqrt(x^2+y^2)" The answer should be 2pi(2-sqrt(2))

My main problem is not knowing how to get started.

Homework Equations

With the example given, it seems we need to find cos(v) first using the equation cos(v) = n*.k/|n|.

The Attempt at a Solution

I found the normal line to be 2xi+2yj+2zk. Using the above formula, I eventually reached the conclusion that z/sqrt(r^2+z^2). I don't know how to use this in an integral and it doesn't follow the example our professor gave us either. Can anyone help?

 

[bigplanet401]

Tame this problem by writing:

z = r cos (theta) (here theta is the zenith)

and

r^2 = x^2 + y^2 + z^2.

Work in spherical coordinates. It'll be that much easier.