Single Application Trapezoidal Rule Derivation

In this segment, we will derive the single application trapezoidal rule. Trapezoidal rules are rules a way to integrate a function from a to b f of x dx. So, what you're basically trying to do is, if we look at the exact definition of an integral, what we are trying to do is we are trying to find an estimate of the area under the curve. we are trying to find the estimate of this area right here. This is the exact area under the curve, or the exact s value of the integral here. So, let's, so what, in trapezoidal rule you basically do is you approximate the function by a straight line. So, let's go and see what that means.

So, in trapezoid rule what you’re going to do is you have this integral which you are trying to calculate from a to b, and you're going to approximate the function by a straight line. So, this is your original function f of x, let's say, and this is your straight line, I’ll call it f sub 1 x. One standing for, it's a first order polynomial or a straight line. And we'll find the area under the straight line and that makes sense, why does it make sense? Because it’s very easy to find the area under a straight line here, because in the shape of a trapezoid and everybody knows how to find the area of a trapezoid I suppose. So what we have basically is that we have this integral a to b f of x dx approximated by the area under the curve f sub 1 x. So, we are approximating the function f of x by f 1 x, and now what we have for, what we have this area as, is the area of a trapezoid, which is given as half of the sum of the length of the parallel sides, so these two marks are parallel sides, multiplied by the perpendicular distance between the parallel sides. And what you’re going to get, is you're going to get one half… Now what is the sum of the length of the parallel sides? If you look at this point right here that's a comma f sub a. That is the coordinate of that, and the coordinate of this point right here is b comma f sub b. So, you can very well see the length of this side here is f sub a. The length of this parallel side here is f sub b, and what is the distance between the two parallel sides? If you find the distance between these two parallel sides, will be nothing but b minus a. So, what we can do is we can write this as b minus a times f sub a, plus f sub b and divided by 2. And that is your single application trapezoidal rule formula.

But what I want to do is I also want to give you a flavor of that if you remember your integral calculus, you must have seen this formula, which is basically saying that the mean of a function from point a to point b is given by integrating the function from a to b and dividing by the width of the interval which is b minus a. So, if I rewrite this formula by doing something like this, you can see the parallel between the trapezoidal rule formula and what I have written here. This is the exact value of the integral. But you can very well see that in the trapezoidal rule, what we have is that we approximated by just calculating the value of the function at two points, and not infinite points but only two points. So, you can very well see that that gives us an estimate of an integral but, of course an approximate one. This is very important to note because that's what most of the integration rules are based on, is that hey it is some width of the interval multiplied by some average value of the function, and that is the end of this segment.