How do I compute the following integral over a submanifold?

If $M\subset \mathbb R^n$ is a $k$-dimensional submanifold and $\phi:\Omega\to M$, $\Omega\subset\mathbb R^k$ a parametrization then the integral of some function $f:M\to\mathbb R$ is defined by $\int_M f=\int_\Omega (f\circ\phi)\cdot \sqrt{\det(D\phi^T D\phi)}$.

In general $D\phi^T D\phi$ is a $k\times k$ matrix. If $k=1$, then $D\phi^T D\phi=[\phi'^T\phi']=[|\phi'|^2]$, which is a $1\times1$ matrix and $\sqrt{\det(D\phi^T D\phi)}=|\phi'|$. This yields the usual formula for the line integral along some curve $\gamma:[a,b]\to\mathbb R^n$ : $\int_\gamma f=\int_a^bf(\gamma(t))\cdot|\gamma'(t)|dt.$

For example the first part of your integral is $\int_{M_1}f(\phi_1(t)\cdot \sqrt{g_{\phi_1} (t)}dt=\int_0^2f(t,\frac t 2)\cdot|(1,\frac 12)|dt=\int_0^2\frac 12 t^2\cdot\frac {\sqrt {5}} {2}dt$.

Hope this helps.