Alternative use of simpson's

Question

I have been brushing up on my calculus. And Im curious as to wether you can use simpsons to find the time instead of distance. since in a give problem its always the time and list of velocities given. Is it possible to use simpsons to get the Time instead when only a set of differing Speed equally spaced Distance traveled is given?

Answer 1

Yeah, you can do that. In terms of Riemann sums, the time taken to run $\Delta x$ at a rate of $V$ is $\Delta t := \frac{1}{V} \Delta x$. Assuming we have $V$ as a function of $x$ (i.e. we know the velocity at any given point, rather than any given time), then the integral $\int_0^M \frac{1}{V(x)} \, \mathrm{d}x$ will determine the time taken to run from $0$ to $M$. You can then apply Simpson's rule to obtain a reasonable estimate of the integral.

Warning: it is assumed here that the velocity is always positive! Otherwise it will be impossible to express $V$ as a function of $x$.