It would help to know the function used to fit the curve (for example nlinfit).
The nonlinear curve-fitting functions require that a model function be provided in order to estimate the parameters. That function can then be used with the integral function to find the area under the fitted curve, or to calculate the fitted values to use with trapz —
t = linspace(0, 5);
s = t.^2 + randn(size(t));
fcn = @(b,x) b(1) .* x.^b(2);
B0 = rand(2,1);
B = nlinfit(t, s, fcn ,B0)
AUC1 = integral(@(x)fcn(B,x), min(t), max(t)) % Use 'integral'
AUC2 = trapz(t, fcn(B,t)) % Use 'trapz'
figure
plot(t, s, '.')
hold on
plot(t, fcn(B,t), '-r')
hold off
grid
Experiment to get different results.
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