Consider $N$ discrete signals $x_1(n), x_2(n), \ldots,x_N(n)$ each a bounded support of size $M$.
To convolve them, we can zero-pad each of them, multiply their FFTs, and take the result's inverse FFT.
I believe this requires taking $N$ FFTs of length $N(M-1)+1$ each, taking $O(MN\,\log(MN))$ time.
However, what I need isn't the full convolution. I only need the area under the convolution above zero: $$\sum_{k=0}^\infty y(n)$$ assuming $y$ is the convolution of our signals: $$y = x_1 * x_2 * \ldots * x_N$$
Is there any algorithm that allows me to compute this value with a better time complexity than that required for computing the full convolution, or are the problems equally difficult?