## Changing the Limits of Integration

It is possible to change the limits of a definite integral to any limits you want by using a linear u-substitution.  To go from the integral from a to b to the integral from a’ to b’, use a u-substitution of u = mx + c, with m = (b’ – a’)/(b – a) and c = (a’b – b’a)/(b-a) with du = mdx.  This technique is used in Gaussian Quadrature for instance.

This is simply a change of scale and a translation.  I was interested in what it looked like as the shape got deformed, always with the same area, to the targeted limits.  Here we ‘move’ a parabola from [3,6] limits to [-1,1] limits:  Moving Parabola to Adjust Integration Limits Here is the final frame:

Parabola now shifted to [-1,1]

Slightly more interesting was to watch this low three peaked object change to a narrow three peaked shape.  The scaling does not change the number of periods displayed because although the period changes, the plotting interval changes accordingly.

Three Peak Beginning Shape

Three Peak Ending Shape

You can watch the video here. That’s all.  I just used this exercise to learn how to create animations using Python.