Efficient root finding algorithm for monotonic function

This is my first time asking a question here, so I may not be asking this in the right place. I am trying to find the roots of a monotonic function with as few function evaluations as possible.

I have approximated a manifold with a piece-wise defined polynomial. The manifold is periodic and so I am only considering its unit cell (one period). I split the unit cell, in general a parallelogram but in this case a square, into triangles.

I then approximate each sheet of the manifold within each triangle with a unique quadratic polynomial.

I would like to find the root that satisfies the equation
begin{equation}
sum_{i}^{mathrm{sheets}} sum_{j}^{mathrm{triangles}} int p_{i,j}(x,y) , mathrm{d}C_{i,j} – A = 0
end{equation}
where $i$ is a sum over the sheets of the manifold, $j$ is a sum over the triangular tiles, $p_{i,j}$ is the second degree polynomial approximation of the manifold’s $i$th sheet within the $j$th tile, and $C_{i,j}$ is the region within a level curve of the polynomial approximation of the manifold’s $i$th sheet within the $j$th tile. Here is a plot of the $C_{i,j}$ for each triangle and sheet for some estimate of the root.

Said another way, I would like to find an isovalue where the area within the level curves of the polynomials, regions where the polynomials are less than the isovalue, is some predetermined value $A$.

At the moment I am using the bisection method, which is very slow because at each iteration it takes a significant amount of time to interpolate the manifold and then calculate the level curves and their containing areas. I may have hundreds of triangles and tens of sheets.

I also tried the regular falsi method but ran into cases where its convergence was worse than the bisection method.