Vectors are not just lists of numbers, they are a mathematical entity. Some operations on vectors are applied individually to each element, others work on the vector as a whole. And some are simply not allowed.
In particular, you cannot use a vector as a limit for an integral. The limit must be a number. If a function uses an argument as an integration limit, only scalar values can be passed for that argument. To evaluate the function at a series of values stored in a vector you must explicity vectorize the function call. That causes the function to be evaluated separately for each element of the vector rather than for the vector as a single value.
Integrals are comparatively slow. They involve multiple evaluations of the integrand in order to estimate the actual integral. If the integrand is somethat that is slow to evaluate (say, an integral) this can become very noticeable. Try to nest integrals a few times and things quickly get unbearably slow. Even with fast machines.
When you want te evaluate an integral for a large number of values of its limit, it ismuch faster to set it up as a differential equation and then solve that. For multiple integrals you can set up a system of differential equations and get results quickly.
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� � � � Tom Gutman
