The instructions for the integrals in Exercises 1–10 have thre...

Question

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.

III. Using Simpson's Rule

a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.

∫ from -2 to 0 of (x² - 1) dx

Accepted Answer

Welcome back everyone. In this problem we want to approximate the integral of x2 minus 4 with respect to x between the limits of 0 and 2 using Simpson's rule with n equals 4 subintervals. Then we want to determine an upper bound for the error ES. Now, before we can approximate the integral using Simpson's rule, we'll need to partition our interval into sub-intervals. So in other words, we're partitioning. Or interval from 0 to 2, OK. Into n equals 4 subintervals. Now remember, the width of each subinterval h will be equal to the difference between our limits divided by the number of subintervals, so that's 2 minus 0 divided by 4, which equals 0.5. So each point X will be equal to 0 plus 0.5 I. Where I ranges from 0 all the way up to 4. Now, let's go ahead and compute the value F of X at each value of X, OK, to help us or that we can use in Simpson's rule. No computing f x for starters, when X sorry, when i equals 0, then we'll be finding f of x0, which will be equal to F0, which is -4. Next, When I equals 1, that means we'll be finding F of X1, which is really f of 0.5, 0 plus 0.5 multiplied by 1. And that is going to be equal to 0.5 square, which is 0.25 minus 4, which is -3.75. So I think at this point you get the idea of what we're trying to do here, right? So for example, for F of X2, it will be F1, and when we substitute 1 into our function, we get -3. Next, we need F X3, which will be F 1.5, which is -1.75. And then we need F of X4, which is going to be F of 2 or 0. So now that we have all of our values, we can apply Simpson's rule. What does Simpson's rule say? Well, by Simpson's rule. OK. By Simpson's rule, then our integral between 0 and 2 of x2 minus 4 with respect to x will be approximately equal to h divided by 3 multiplied. By FF X not. Yeah Plus 4 multiplied by F of X1, + 2 multiplied by F of X2 + 4 multiplied by F of X3. Plus F of X4, OK. Now when we do that, we'll end up with 0.5 divided by 3 multiplied by -4 plus 4 multiplied by -3.75. Plus 2 multiplied by -3 + 4 multiplied by -1.75 + 0. I know when we simplify here, we get this to be -16/3, OK, so this is our approximation based on Simpson's rule. Now we can move on to the next part of our question. What will be an upper bound for our error of magnitude ES? Well, to solve that one, OK, recall that the errorborne formula for Simpson's rule basically tells us that the absolute value of our error is less than or equal to b minus a 5th, OK, divided by 180 and 4th, multiplied by the maximum value on the interval from A to B, OK. Of the absolute value of the 4th derivative of f x. Now we have all the inform, most of the information that we need. We know B and A are upper and lower limits. We known or the number of subintervals, which is 4. All we need to do now is to find a 4th derivative of F of X and use it to help us to find the maximum value of that 4th derivative on the interval. Now, to find the 4th derivative, we can just differentiate FX 4 times. Given F of X is x2 minus 4, then our first derivative will be 2 x. The 2nd derivative will be 2. Then the third derivative will be 0. Which means our 4th derivative will also be 0. I know if our 4th derivative is 0, that means the maximum value is 0, OK, so the maximum value on the interval of the absolute value of the 4th derivative of FFX. is going to be 0 and that implies then that the absolute value of our error is going to be less than or equal to 0. Hence, the Simpson's rule approximation is -16/3 with an upper bound for the error of 0. Thanks for watching everyone. I hope this video helped.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.III. Using Simpson's Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ES|.∫ from -2 to 0 of (x² - 1) dx

Watch next

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from -2 to 0 of (x² - 1) dx