Just numeric problems.
You are correct that Mathcad cannot integrate piecewise defined functions as well as it can analytic functions. The algortithm for numeric integration assume continuous derivatives, through at least two derivatives, and generally require bounded second derivatives. Discontinuities, even just corners, don't work all that well. Generally if you have a function defined piecewise (and the joins are not completely smooth) it is best to break up integrals and calculate separate integrals for each of the ranges.
Here the problem is the implied sampling (true any numeric algorithm -- the integrand can only be evaluated at a finite number of discrete points) and the relationship of the sampling points to the sharp peak. Because that peak contributes so much to the integral, the effect is quite noticeable. Note that the approximation as a sum produces the same sort of jagged result.
You can improve the integral considerably by using a smaller value of TOL. At 1E-9 the curve is fairly smooth. Similarly you can increase the quality of the Euler sum by increasing the number of intervals. 300 (rather than 30) gives a reasonable looking answer).
But calculating a whole lot if integrals this way (varying the upper limit) is a slow, and not all that accurate, way of doing the calculation. It is much faster, and often more accurate, to set it up as a differential equation and then use one of the ODE solvers (I'm partial to Rkadapt, myself). If you nose about the collaboratory you should be able to find some expamples of doing this.
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� � � � Tom Gutman
