Find the surface area of the portion of the sphere $x^2 + y^2 + z^2 =3c^2$ within the paraboloid $2cz =x^2+y^2$ using spherical coordinates. ($c$ is a positive constant)
I've found it in cartesian coordinates and then polar coordinates by taking limits $r$ from $0$ to $\sqrt 2 c$ and $\theta$ from $0$ to $2\pi$ and got the answer $4\sqrt 3\pi c^3$. But for the spherical coordinates I am getting the limits $\phi$ from $\arccos(\sqrt 3)$ to $\pi/2$ and $\theta$ from $2\pi$ to $0$ and the answer is $6\pi/\sqrt3 c^2$ which is $2c$ times less than in cartesian coordinates. What am I doing wrong?