You can probably apply a highpass filter to the data to isolate the peaks. This should remove the low frequency/offset of the plot, while allowing the quickly changing peaks to pass through unchanged.
Alternatively, you can locate the peaks of the data with something like:
[pks, locs] = findpeaks(data);
Because the peaks seem to have a consistent width, you can divide the data into small "subsections" and plot each individual subsection.
To find the area under the curve, you can use a trapezoidal approximation using one of the following:
cumtrapz();
trapz();
