Surface Area on x^2+y^2+z^2=4x, inside x=y^2+z^2

[Daniel_Feldman]

I need to find the area of the surface that lies on the sphere x^2+y^2+z^2=4x and inside the paraboloid x=y^2+z^2.

So I found the partial derivatives and then put them in the integrand in terms of polar coordinates. My outer limits will be 0 to 2pi, but I do not know what my inner limits will be. Is this easier to do in, say spherical coordinates?

 

[Unco]

Spherical coordinates will simplify things. Restating the problem as the surface of the sphere x^2 + y^2 + z^2 = 4 inside the parabaloid z + 2 = x^2 + y^2 (by translating along the x-axis 2 units and switching x and z axes), gives straightforward limits for theta: 0 to 2pi and phi: -pi/3 to pi/3, and a familiar integrand.