How to compute the total curvature?

Question

I have the surface $M\subseteq\mathbb{R}^3$ parametrized by $\sigma(s,\theta)=(\varphi(s)\cos{\theta},\varphi(s)\sin{\theta},\kappa(s))$, where the functions $\varphi,\kappa$ satisfies $$ \left(\partial_s\varphi\right)^2+\left(\partial_s\kappa\right)^2=\varphi^2$$ $$ \partial_s\kappa = \tau^2+\varphi^2$$ and $\tau\in[0,\frac{1}{2})$ is constant, $\varphi(0)<\tau,\ $ $\partial_s\varphi(0)=0$ and $\kappa(0)=0$. Iwant to calculate the total curvature of $M$. My try was: the first fundamental form is $$ I = \varphi^2I_{2\times2}$$ where $I_{2\times2}$ is the identity matrix of dimension $2$. So, $M$ is conformally parameterized surface, so we can apply the formula $$k = -\frac{1}{\varphi^2}\Delta\log{\varphi^2}=\frac{\varphi''\varphi-(\varphi')^2}{\varphi^4}$$ But this lead to nothing useful :(. Any ideas?