It is critical to know how the peak heights and sds were calculated. (I take "mean" in the question to be a mistaken way of referring to a height. Without the heights, the problem is hopeless; it would be like requesting a formula for the area of a rectangle given only its width and location.)
One would expect, as Joris Meys' answer and its commentary suggest, that the area could be estimated as a sum of Gaussians. Actually, we don't need to assume a Gaussian shape; almost any standard (preferably unimodal, continuous) shape will do, because the area will be proportional to the peak height (a y-scale factor) and the sd (an x-scale factor), whence the total estimated area ought to be a constant times the sum of height*SD and the relative contribution of each peak will equal its height*SD divided by this sum. But this all assumes the heights and sds were fit to the curve with such an application in mind.
I realize there are many problems with such a formula, but let's not get carried away by all the detail in the example graph: the problem as posed says that the "means" and SDs are the only information available.