Text description - screencast for interactive 4

[In this video, an electronic x and y graph is used to show how equations 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. Below the line, a series of shaded rectangles run along the x-axis, their top left corners touch the line. As the values in the equation are changed, the shape of the graph changes and also figures marked "Area of rectangles", "Total area of rectangles and "Actual area under graph" change. The narrator describes the changes.]

NARRATOR: In this interactive, we explore the left endpoint estimate. We're going to look at the left endpoint estimate under the graph of y = x squared + 1 from x = 0 to x = 5. So at the moment we see that n is set to 5, that is, that we're using five rectangles to estimate this area. So given that we're looking at the area from 0 to 5, five rectangles, each have a width of 1 and we can calculate the area of each of those five rectangles. You'll see in this interactive over here where we have 'Sequence', that's giving us the x coordinates of each of the boundaries between the rectangles.

The 'Areas of Rectangles' is, obviously, calculating the area of each rectangle, so that is, that x value from that sequence at the left endpoint - so in this case using the x values of 0, 1, 2, 3 and 4 - and multiplying them by the heights of the rectangles at those corresponding positions. We'll see over here on the left we're being told the n value and also delta x. Delta x refers to the width of each rectangle, which in this case is 1.

We're also being shown the total area of the rectangles along with what the actual area under the curve of y = x squared + 1 between 0 and 5 is to allow us to make a comparison. So we see that as we increase the value of n, we're getting more rectangles, obviously. Those rectangles are each getting thinner. So each rectangle now has a width, as we can see down here, of 5 divided by 7 - we've divided a distance of 5 into 7 equal units.

And we're also seeing each of those x values that creates the boundaries between each rectangle. So the first one is, obviously, at five-sevenths, the next one is at 2 times five-sevenths, the next one is at 3 times five-sevenths, etc. So we're getting more rectangles.

We also note that as we increase the number of rectangles, the total area of the rectangles is getting closer to the actual area. We can see that quite physically in the diagram above and we can also see that in the calculations that are occurring down here below. So as we watch the number of rectangles increase, we physically see that area get close to the actual area and we also see numerically the area of the left endpoint estimate get closer and closer and closer to the actual area.