We have previously seen how we can approximate the integrals of functions by noting that they are essentially the areas under their graphs and so by drawing simple shapes upon those graphs and adding up their areas we can numerically calculate the integrals of functions that we might struggle to solve mathematically.

Specifically, if we join two points

In regions where a function is rapidly changing we need a lot of trapeziums to accurately approximate its integral. If we restrict ourselves to trapeziums of equal width, as we have done so far, this means that we might spend far too much effort putting trapeziums in regions where a function changes slowly if it also has regions where it changes quickly.

The obvious solution to this is, of course, to use trapeziums of

Specifically, if we join two points

*x*_{1}and*x*_{2}on the graph of a function*f*with a straight line and another two vertically from them to the*x*axis then we've drawn a trapezium.In regions where a function is rapidly changing we need a lot of trapeziums to accurately approximate its integral. If we restrict ourselves to trapeziums of equal width, as we have done so far, this means that we might spend far too much effort putting trapeziums in regions where a function changes slowly if it also has regions where it changes quickly.

The obvious solution to this is, of course, to use trapeziums of

*different*widths.Full text...