Text description - screencast for interactive 3

[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. A series of shaded rectangles run along the x-axis, the line crosses about halfway along their top sides. 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 midpoint estimate. Here, we're looking at the graph of y = x squared + 1 and attempting to find an estimate for the area under this curve between x = 0 and x = 5. At the moment, n is set to 5. That is, we've divided this region into five rectangles. That then makes the width of each rectangle, which is delta x down here, equal to 1, that is, 5 divided by five, and each of the boundaries of the rectangles can also be obtained using the width of this rectangle.

So the first boundary is at x = 1. The next boundary is at 2 times the width of the rectangles. That is, x = 2. The next boundary is at 3 times the width of the rectangles, that is, x = 3. And as we increase the value of n, we see the number of rectangles increasing. That makes their width decrease so now we have the width of each of the seven rectangles is equal to 5 divided by 7 and we also see the boundaries changing, so the boundary between the first and the second rectangle is now at five-sevenths - that is, x1 is at five-sevenths, x2 is at 2 times five-sevenths, which is ten-sevenths, etc.

And down here we're seeing the areas of each of these rectangles being calculated using the height at the midpoint between each of these boundaries to calculate the area. Down here we're seeing the total area of the rectangles shown and the actual area underneath this graph. And we should see that as we increase the value of n, firstly we see more and more rectangles and we can also see that the area of the rectangles is getting closer and closer and closer to the area of the actual area and we can see that both graphically and also by looking at the numeric calculations down below here.