In this segment, we'll take an example of the single application trapezoidal rule.
So, this is the example we're taking; we're asked to integrate a function which looks like this.
We’re asked to use the single application trapezoidal rule find the true error and find
the absolute relative true error. So, let's go and see how we can go about doing that.
So, we have this integral given to us.
We already know that the formula for the single application trapezoidal rule
is given as follows. The general case,
so in this case we have
the values of a and b: 1.3 and 0.1. And we're going to calculate the value of the function at
0.1 and calculate the value at 1.3, and we’ll take the average value of those two. So, this gives us
1.2 and this one is 5, 0.1 times e to the power minus 2 times 0.1, plus 5,
1.3 e to the power minus 2 times 1.3,
divided by 2. And this value here turns out to be as follows once you do the calculations.
So, this is what we get by using the single application trapezoidal rule
as the value of the integral. Now let's go, this was part a, let’s go to part b
where we have to find the true error as well as the relative true error.
To find the true error, we need the true value and the approximate value which
we just calculated in the previous part. The true value is given to us as 0.89387,
and if you are interested in, you can find the true value by just doing integration by parts.
And approximately, we found out from the previous part was this and this number here turns out to be
0.35858. Now if we want to find the absolute relative true error, in that case, it will be
the true error divided by the true value absolute value of that, and we calculate in terms of
percentage we multiply by 100. So here we get this to be this quantity here, which we just found out.
The true value is
this quantity, and we multiply 100 and we get a relative true error of 40.115%, which is huge,
but you gotta understand this is only a single application trapezoidal rule. It’s a little bit
out of the ordinary because of the way the function behaves; because if you look at the
way the function behaves. It’s like this, this is your 0.1, and that’s your 1.3. As we can see,
there's a lot of true error here and that's the approximate value, the shaded area here like with
the crosses, and this one is the true error. So, that's why I was saying that in this example here,
it's a little bit inflated. I shouldn’t say, call it, inflated but it is and out of the
ordinary example that we're getting this much true error. And that's the end of this segment.