Gaussian curvature of a surface of revolution
= Gaussian curvature of a surface of revolution
{c}
For the parametrization
$$
X(x,\theta)=(x,f(x)\cos\theta,f(x)\sin\theta),
$$
the Gaussian curvature and area element are
$$
K=-\frac{f''}{f(1+f'^2)^2},
\qquad
dA=f\sqrt{1+f'^2}\,dx\,d\theta.
$$