If i am using a Riemannian metric on the cone $C$ with conical point at origin and metric given by $dt^2 +t^2d\phi^2$ with $t>0$ and $\phi\in (0,2\pi )$ how to use this to do an integral. For example, for the function $f(t,\phi)=t^{\frac{1}{2}\pm \sqrt{\frac{1}{4}+\frac{k^2}{a}}}\cos\phi$ i want to find conditions such that $f\in L^2(C)$. So, my attempt is
$$f\in L^2\leftrightarrow \int_{0}^{\infty}\int_{0}^{2\pi}|f(t,\phi)|^2 dtd\phi$$
but how can i use the terms $dt^2 +t^2d\phi^2$ on the integral above? I mean if i applied power two in the integrand then $1\pm2 \sqrt{\frac{1}{4}+\frac{k^2}{a}}\cos^2\phi$ but $\cos\phi$ is bounded and the important term is $t^{1+2\pm \sqrt{\frac{1}{4}+\frac{k^2}{a}}}$ but how to associate with the coefficient $1$ of $dt^2$ and the coefficient $t^2$ of $d\phi^2$? I know that the matrix that represent the metric $g$ is \begin{array}{} 1\quad0\\0\quad t^2 \end{array} But to be honest i do know how we use it when we need to do an integral. Please, somebody can help me? or give me one reference to read with examples.