Text description - screencast for interactive 1

[In this video, an electronic x and y graph is used to show an equation can be presented in graph form. Below the graph, adjustable sliders allow individual values in the equation to be changed. On the graph, a line curves upwards from the x-axis. The area below it is shaded. As the values in the equation are changed, figures labelled "Areas of trapezia" "Total area of trapezia" and "Actual area under graph" change. The narrator describes the changes.]

NARRATOR: In this interactive, we explore the trapezoidal estimate. Here we're looking at the graph of y = x squared + 1 and attempting to approximate the area underneath this graph between x = 0 and x = 5 and we're doing this using the trapezoidal estimate, which involves creating a number of trapezia to approximate the area.

Currently n is set to 5, which means we've divided the interval into five equal width trapezia. Each trapezium has a width of 1, as we see. As we increase n, we're obviously increasing the number of trapezia, hence decreasing the width of each trapezia and also seeing that the area estimated by the trapezia is in fact getting closer and closer to the area of the curve. This is becoming quite difficult to see quite quickly on the graph given that the trapezia creates such a good approximation, but we can certainly see from the numerical calculations in the bottom left that the total area of the trapezia is indeed getting closer and closer to the actual area underneath the graph. This approximation, that is the trapezoidal estimate, is a much better approximation than either the right endpoint or the left endpoint or in fact the midpoint estimate.

In fact, the trapezoidal estimate is the average of the left endpoint and the right endpoint estimates and you'll have noted if you've already had a look at those interactives that the left endpoint estimate gives an approximation that's slightly underneath the actual area and the right endpoint estimate gives an approximation that's slightly above the actual area and so an average of those two areas is much more accurate and the average of those two areas is in fact equal to this trapezoidal estimate.