Text description - screencast for exercise 4

[The narrator reads out the onscreen text.]

NARRATOR: Exercise 4. Show that the trapezoidal estimate is the average of the left endpoint estimate and the right endpoint estimate.

[An x and y graph features an upward sloping parabola. The equation y equals f of x is written nearby. Beneath the line is a series of trapezia, their top left corners touching the parabola, so each trapezia is taller than the last. The shortest rectangle is marked with the coordinates (x0, f of x0), the tallest rectangle is marked with the coordinates (x n minus 1, f of x n minus 1)]

NARRATOR: First we consider the left endpoint estimate, which I'll abbreviate to LEE. And we note that the width of each rectangle here is the difference between consecutive x values, so, for example, the width of that first rectangle is x1 - x0, the width of the second rectangle is x2 - x1, the width of the third rectangle x3 - x2, etc. So we can generalise that as (xi + 1) - xi. And we're going to call that width of each rectangle 'delta-x', just to make life easier. So the left endpoint estimate can be calculated by adding up the areas of all of these rectangles. And we note that with the left endpoint estimate, the first rectangle has a height equivalent to the value of the function at x0. So our first rectangle has an area of f(x0) multiplied by delta-x. And the next rectangle has an area of f(x1) multiplied by delta-x. And we can continue right up to the last rectangle which has a height equivalent to the value of the function at (xn-1). So the final rectangle has an area of f(xn-1) multiplied by delta-x.

Next, thinking about the right endpoint estimate, which I'll abbreviate to REE. Again, we know that the width of each of these rectangles - we're going to refer to that as delta-x - and that's equivalent to the difference between consecutive x values. And here we see the main difference with the right endpoint estimate is that the height of the first rectangle isn't the value of the function at x0 but instead the value of the function at x1. And so the area of the first rectangle is f(x1) multiplied by delta-x. The area of the next rectangle is f(x2) multiplied by delta-x. And we can continue right up to the final rectangle, which has a height equivalent to the value of the function at xn. So the final rectangle has an area of f(xn) multiplied by delta-x. And we can factor out the delta-x which is common to each of these expressions and see that the right endpoint estimate is just equal to the sum of the values of the function everywhere from x1 up to xn multiplied by delta-x.

So now with our expressions for the left endpoint estimate and the right endpoint estimate, we can work out an expression for the average of those two estimates. So we would add them together and divide them by two to calculate the average.

So next I've just factored out the delta-x from both expressions and also rewritten the division of 2 as a multiplication by a half. And now we can see that interspersing the terms from the left endpoint estimate with the terms from the right endpoint estimate, that we're getting f(x0) + f(x1) and then + f(x1) + f(x2) + f(x2) + f(x3). And we can continue these little pairings, which when we split them all up again, so we take each of those little pairs and break it out with a half and a delta-x as a factor, we see we get these little sums. So we get half times f(x0) + f(x1) then multiplied by delta-x, + half times f(x1) + f(x2) multiplied by delta-x.

And this may seem trivial, but what we should be able to note is when we actually think about the trapezoidal estimate, is in fact each of these expressions is how we calculate the area of each of these little trapezia. So that first trapezium in this diagram here we see has a height, if you like, so that's the perpendicular height between the two different-length parallel sides. That's delta-x. And we also know that to find the area of a trapezium, we're finding half multiplied by the sum of the length of the two parallel sides and then multiplied by that height that I referred to before. So the area of the first trapezium would be found by doing half times the value of the function at x0 + the value of the function at x1, and then multiplying that by delta-x. And we can repeat that all the way up to our final trapezium as shown in that equation there. So we can see that simply by manipulating the algebra of what's going on behind the left endpoint estimate and the right endpoint estimate, we can show that in fact the average of the left and right endpoint estimates is equivalent to the trapezoidal estimate.