In this problem. We have an hourly temperature data for Duluth over a 12 hour period, and we want to fight an accurate approximation to the average temperature over T still far, period. Knowing that the ever stay emptory given as 1/12 times the integral from Syria to 12 off the champion to function and the temperature function is considered a continuous function over these closed interval. So the integral definite into rule he's well defined and the temperature data is hearing at each hours turning zero and up to 12. So we can't use and escape the Simpson rule, which give us the most occurred, um, approximation to the definite into Brooke we have of the method. We have seen this chapter. So we apply. We're gonna apply that within Delta T off 12 minus zero, which is the length off the time interval over the number of 70 rules, which are 12. That gives us a delta t of one, and the notes in times are TF, TKT or DISIP cable syrup plus K times Delta T. That is one. So we have peace of Gables key for K from zero 2 12 So the nodes of time are Syria want to, and so are up to 12th in which we have the temperatures even in the table, so that we can ride this Simpson rule and we have to. That is as of 12. Equal to temperature at zero plus four times Demeter at one plus two times demeter it too close, full time. Still. Picture at three plus two. Temperature of four before 10 35 plus two Tim direct six plus four temperature at seven. Plus two temperature at eight, plus four temperature at nine plus two tempter and 10 plus four tempter and 11 plus Demeter. 12 Old doesn't apply by Delta T over three, which is able to 1/3 in this case. So this is equal to T. We are going todo zero and then we see that are there are terms that normal by before his wand. It's one here. Here is one here and he's one. There are all the earth's there, moved to our multiply by two so we can affect your hope. Number four here. Sorry. With four times t at one bless T and three plus t five. First he had 67 Sir. Rusty at nine plus De uh, 11 plus two times t A tube list yet War plus de at six plus t at eight plus de death. Then last year, empty of 12. Well, that multiply by Want, sir? Now we go. We're gonna replace the values from the tables here. So we get 09 glass Whole times 11 plus 12 plus 15 plus 19 plus 22 less 24 close to times 11 plus 14 17 plus 20 plus 24 plus de a 12 is 25 can. Okay, And that multiplied by 1/3. Which is it going to divide that some that is fresh by three. And so he's in a calculator. We find that this is sick legal to 206. So the integral from Syria to 12 off the temperature function, he said recently. Well, too, 206. And then, with that, we can say that the average timid temperature in on the 12 hour period for the Duluth function a city he separates timidly with 100 and 6/12. Give us, um, 17 point to, but he's the average of Team two rece every seeming to equal to 17 going to. We can say that the average temperature over the 12th our period poor to lose city. He's about its approximately well too 17 point to this is final.