If you have met integration before, and even if you are really comfortable integrating all sorts of complicated functions, it is still possible that you haven’t spent much time thinking about the relationship between numerical and analytical integration, or wondered why there is more than one method/algorithm for numerical integration. If so, here’s a video just for you…
The green (underestimation) and red (overestimation) are really interestingly compared at http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-intassum-2009-1.pdf . We said in the video that one consistently overestimates and one underestimates the area, so that the real area must be between the two. In the link provided, the over- and under- estimates are quantified, and evaluated in the limit of [latex]\Delta t \rightarrow 0[/latex]. As the under- and over-estimates shrink, the boundaries within which the ‘real’ value is contained get ‘squeezed’ until the ‘real’ value is arrived at.
Note also that the ‘green approach’ won’t always result in underestimation – it depends on the shape of the graph (and likewise for the ‘red approach’). You can see this in the golf ball example of ‘impulse’ in Section 3 – the first half of the area is underestimated and the second half is overestimated.
You can investigate all this in the following interactive graph, which you may have seen before in Section 03. This time, change ‘how they operate’ in the ‘Sum type’ box. Notice how the different sums converge to the same value as you increase the number of rectangles, regardless of whether you select ‘left’, ‘right’, ‘midpoint’, etc (note that you may have to drag the slider to update the graph once you have changed the sum type).