29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rule...

Question

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

Accepted Answer

Welcome back everyone. In this problem, we want to use the trapezoid rule to approximate the integral between 0 and a half of pi of the cosine of 2 X multiplied by the sine of 4 X with respect to X using N equals 12 subintervals. Then we want to calculate the absolute error where the exact value of the integral is 2/3. Now, if we're going to use the trapezoid rule to approximate the integral, let's first ask ourselves, what do we know about the trapezoid rule? Well, recall, OK, that the composite trapezoid rule. OK. On the interval from A to B, OK. With an equal sub intervals. Alpha with Delta X. Is going to be equal to half of delta X multiplied by F of X0, OK, plus 2 multiplied by the sum from I equals 1 to N minus 1 of F of XI, OK, plus F of XN. So what does this tell us? Well, for us to use the trapezoid rule to approximate, we need the width of each subinterval delta X. We'll need to figure out each point Xi, OK? And then we'll also need to evaluate our points, evaluate our function F at the points XI. So first, let's start with our sub interval, width of our subintervals, OK? Now, we know. That given A equals 0, B equals a half of pi, N equals 12, OK. And F of X being equal to 2 X 4 X based on all the information that we're given in our problem. Then the width of Y subinterval delta X is gonna be equal to 0.2 of pi minus 0 divided by 12. That would be equal to a half of pi divided by 12 or 1/24th of pi. Next, we also know that Xi then is gonna be equal to I pi divided by 24, OK, where I goes from 0 all the way up to 12. So now that we have the width of our sub-interval and XI, then we can use that to evaluate our function at the values of XI. So really, we're evaluating F from X0 to X12. Let's set up a table to help us organize that a bit better, OK? So, let's set up a table with XI and F of XI. I know we have that and let's extend our table this way. And now, starting with 0, OK, then we're going to go to X1, which is. 1 24th of pie. Then we're gonna go to a 12th of pie. And remember, the width is 124th of pie, so we're going up by that amount each time. Next we're gonna have 1/8 of pie. And then a 6 stuffed pie. OK. And then we're going to have 5 24th of pie. And then we're gonna have a 4th off by. OK. And then we are going to go to. 74724 soft pie, sorry. And then a third of pie. Now if I go ahead and let me extend this table, OK. Let's continue this table below. Then we come back, we have XI and F of XI. Now if we extend this table here. So we write that right here, yeah. Then continuing, we're going to have uh 3/8 of pie. OK, uh, 512 soft pie. OK, 1124 of pie. And then finally, X12, which is a half of pi, OK? So these are going to be all of our values for XI. And know for us to solve all we need to do is to substitute uh XI into our function. So for example. When Xi equals 0, then we're going to find F of 0. So that's the cosine of 2 multiplied by 0 multiplied by the sine of 4 multiplied by 0, which is going to be equal to 0. When X equals 124th of pi, then it's going to be root 6 plus route 2 divided by 8. When it's a 12th of pi, it's 3/4 when X's. Uh, an 8 of pi, F of X is route 2 divided by 2. When it's a 6th of pi, it's route 3 divided by 4. When it's 5 24th of pi, it's route 6 minus route 2 divided by 8. when it's when it's uh 4th of pi X6, then F is gonna be 0. When it's 7/24 of pi, we get route 6 minus route 2 divided by 8. When it's when it's a third of pi, sorry, we get route 3 divided by 4. When it's 3/8 soft pie, we get route 2 divided by 2. when it's 5/12 soft pie, we get 3/4. When it's 11 24th soft pie, Route 6 plus route 2 divided by 8. And when it's a half of pie, we get 0. So now if you think about it here comparing it to our formula for the trapezoid rule, we have X0 then we have XI from X1 all the way up to N minus 1, which is X 11, and then we have X12, OK, which is XN from our formula. So now we can substitute all our values. For starters, OK, so now, with us, we can say then that TN is gonna be equal to delta X divided by 2 multiplied by F of X0 which we know is 0, plus 2 multiplied by or sum from I equals 1 all the way up to N minus 1, which is gonna be 11 of F of Xi, OK, plus F of X N F X 12, which is also going to be equal to 0. So really then, all we need to do is to find our sum from XI or X1, OK? All the way up to X 11, OK? So this is pretty much all we need. So what are those going to be? Well, when we add all of those up, all of these values here, OK. Then we should get it if we substitute delta X here, we should get it to be pi divided by 24 divided by 2 multiplied by 2 multiplied by 5.0054. OK? That would be our sum to 4 significant figures. And now when we go ahead and solve here two cancers 2, so we're gonna have 1 24th of pi multiplied by 5.0054, which It's going to be equal to 0.6552. So this tells us then that our integral. Therefore, the integral between 0 and 2 of pi of cost 2 X. Multiplied by the sign of 4 X with respect to X is approximately equal to 0.6552. Now that we have that, Ken. All we need to do now is to calculate the absolute error. Now remember to find the absolute error. Absolute error. OK. It means it's going to be equal to the absolute value of the approximation. Minus the exact value. So this is gonna be, oops, it should be an absolute value here. So this is gonna be the absolute value of 0.655 to minus 2/3. And that would be equal to 0.0115. So, in summary, our approximation to the integral is 0.6552 and the absolute error is 0.0115. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Midpoint Rule

Trapezoid Rule

Error Analysis in Numerical Integration

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