I have a line integral over a scalar field in the form of
$$ \int_{t=0}^{t=1} f(x(t),y(t)) \sqrt{(d x/d t)^2 + (d y/d t)^2} d t $$
Function $f$ is
$$ f(x) = 0.5 \exp (-\gamma || x - m_1 ||^2) + 0.5 \exp (-\gamma || x - m_2 ||^2 ) $$
where $x$ is actually $x=[x,y]$ to fit the first equation. The $x(t),y(t)$ are cubic polynomials. I am basically trying to find the area under parametrized cubic curve over a scalar field defined by an RBF function.
However even at polynomial degree 3 and with a simple RBF function as seen above, the symbolic integration gets very hairy. I have to numerically integrate this, I was wondering about the best approaches for that. Quick idea is dividing $t$ into N blocks, and make $dt = 1/N$, get $x,y$ from $t$, do the multiplication, sum, do this $N$ times, we are done.
Is this feasible? Should I use trapezoidal integration, or other methods for this?