CHAPTER 07.03: SIMPSON'S 1/3RD RULE: Integration: Derivation

In this segment, we're going to take an example of Simpson's 1/3 rule of integration. And so we're going to use the formula to see that how we can apply that to find an approximate value of an integral.  So let's suppose the example tells you use Simpson's 1/3 rule to find this integral, 0.1 to 1.3, 5 x e to the power -2 x, dx. So that's what you are asked to do, you are asked to find out what is the approximate value of this integral by using the Simpson's 1/3 rule.  Now, we just derived the Simpson's 1/3 rule as follows, that this is the approximate value of the integral given by Simpson's 1/3 rule, that you have h divided by 3 here, and then it is multiplied by the value of the function at a, plus 4 times the value of the function at a plus b, divided by 2, plus the value of the function at b, that's what we get as the approximate value of the integral by using Simpson's 1/3 rule.  Again, I'm going to mention that the reason why it's called Simpson's 1/3 is because we have the 3 in the denominator there, and the value of h is simply the half of the interval width, so h is b minus a, divided by 2. So all we have to do is to substitute the values of a and b, find h, find the values of the function at a, the midpoint, and the value at b, and we are done.  So let's go ahead and write this down.  So we have the function itself which we are trying to integrate, or the integrand itself is 5 times x e to the power -2 x, a is given as 0.1, b is given as 1.3, your h will be b minus a, divided by 2, so this turns out to be 0.6. So the value of the integral 0.1 from 1.3, of this f x e to the power -2 x, dx is approximately equal to h, which is 0.6, divided by 3, h divided by 3, times the value of the function at a, so the value of the function at a is . . . the value of a is 0.1, plus 4 times the value of the function at the midpoint, which is at 0.1 plus 1.3, divided by 2, plus the value of the function at 1.3.  So here I get, it is 0.2 times the value of the function at 0.1, plus 4 times the value of the function at 0.7, plus the value of the function at 1.3.  So the way you've got to look at it is that you have to calculate the value of the function at the lower limit of integration, plus 4 times the value of the function at the midpoint between the upper and lower limit of integration, plus the value of the function at the upper limit of integration, so that's what is the approximate value of the integral of this by using Simpson's 1/3 rule.  All I have to do now to substitute the value of the function now at those particular points, I get 0.2 times the value of the function at 0.1, will be 5 times 0.1 e to the power -2 times 0.1, so that's the value of the function at 0.1, plus 4 times the value of the function at 0.7, now plus 1 times the value of the function at 3, that's what we get there.  So if I do some more simplification, I get this number, the value of the function at 0.1 turns out to be 0.4094, the value of the function at 0.7 turns out to be 0.8631, and the value of the function at 1.3 turns out to be 0.4828. So once we have found out the values of the function at these three points which we have to do, the value turns out to be 0.8689. So that's the approximate value of the integral which we are trying to find by using Simpson's 1/3 rule. Now, what is the exact value?  The exact value of the same integral, 0.1 to 1.3 of 5 x e to the power -2 x, dx is equal to 0.8939, that's the exact value up to four significant digits for this particular value of the integral, and I'm going to ask you to do this as homework, and the hint for finding the exact integral such as this one is to do it by parts.  So you can see that this is the approximate value which you are getting by using Simpson's 1/3 rule, and this is the exact value which you are getting by using your integral calculus knowledge. So let's go ahead and look at what the absolute relative approximate error is.  So if I was going to look at the absolute relative approximate error for this particular case, it is the, I'm going to write down the definition of the absolute relative approximate error, it's true value, minus approximate value, divided by true value, times 100, that's what we have.  Now, we already know what the true value is, that's part of your homework, 0.8939, and the approximate value which we obtained by using Simpson's 1/3 rule is as follows, divided by the exact value, which is 0.8939, multiplied by 100, I'm just trying to get the value of the absolute relative approximate error in terms of a percentage, and this turns out to be 2. . . . approximately 2.8 percent, that's what I get as the relative true error for this example.  And that's the end of this segment.