5

8. function f(r) is graphed helow0.51525(a) Estimate f(r) dr by a left Riemann sum with N =6.(h) Estimate f(r)dr by" 4 right Ricqan Sunt, with . = 6....

Question

8. function f(r) is graphed helow0.51525(a) Estimate f(r) dr by a left Riemann sum with N =6.(h) Estimate f(r)dr by" 4 right Ricqan Sunt, with . = 6.

8. function f(r) is graphed helow 0.5 15 25 (a) Estimate f(r) dr by a left Riemann sum with N =6. (h) Estimate f(r)dr by" 4 right Ricqan Sunt, with . = 6.



Answers

In Exercises $15-18,$ use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \text { on } [0,4]$$

Right. Question 120. We're supposed to use the left hand sums for equals 1 10 in 100 or the function y equals X squared, minus four on the interval from 0 to 2 in order to find the average value of each of those. Okay. Supposed to do this on a computer program? So many interval suspicion. Get out here. Okay, So basically, what we want to do is to, um, enter these pieces of information indoor computer program. Um, but the function is the left endpoint. The right end point, the number of divisions you want, we're gonna do 1 10 and 100. Um, and then the choice of method program I use, um, if he put in zero, it's a left limit 21 you get right and limit in some. All right, so here we go. We've got these. And so let's take a peek at what we get when we enter this information, or Yeah, we do that. We get this scraps of yours. This is Ex player minus four. It's the years the vortex down here. Negative for you also see that the, uh, for n equals 10 that, uh, the left hand sums expert and equals anything. Left hand sums are going outside of, um of the function and that the area is negative. That's below the assets. So, um l one, uh, if you do just one division, that would be this whole big rectangle right here, which would be two times negative pours. That's B negative eight. Um, that's what the computer said, too. But we want the average value. So since the length here is to, then we need to divide this by tubes are average values for and then we go on in l 10 which is what this picture of that ends up being negative. Five 0.72 which is us negative. 2.86 So it gets smaller, right? The summits smaller because these rectangles are hitting her better than this whole big chunk. First, when she divided into 100 it basically becomes smooth. Her There's not very much error at all. Yet negative 5.37 for the sum. And then again, when you divide that by two needed to corn 1 65 divided by two because you want the average value of it. Now the actual average value is negative. Eight. Thirty's won't compare this to that. And we're before we had a right hand, some and left, and some in that in between them was the actual. But this problem is showing is that the more divisions you have, the more divisions you have closer. Your answer gets to the actual answer. Um, so see how this way off for the original and equals one. And then as the end gets bigger, you close? Yeah.

Okay. Using our calculator, we determined that l one other words from the left hand side is equivalent to negative eight. L ton is equivalent to negative 5.72 and a 100 is equivalent to negative 5.38 We know the exact solution is negative. Eight over three, which you're in. The one need a calculator toe. Confirm. This is negative 2.67 So, as you can see, L 10 is most accurate compared to the given answer.

In discussion. We are required to find they left remain some for the given graph and we have the interval zero comma aid. And the number of our residents are for So let's see how to solve this question. We know that if F is a continuous function, that left remain some with an equal subsidence for F over the interval mm comma B can Britain is left remain some difficult to submission case equals to zero To any equals to one. F X one bill tax. So this will be calls to F Acts not multiplied by delta X plus effects one multiplied by delta explorers. FX two multiplied by dale tax plus so on. Plus FXN -1 multiplied by dale tax here Fx zero. Fx one fx two and fx and minus one are the value of functions at initial 1st, 2nd and and minus ones of dividends. So now let's find the subdivision or the birth of south division. The formula to calculate worth of sedatives and competitiveness, delta taxes equals two B minus a upon end. Therefore deal tax will be called to 8 0 divided by four. So it will be called to to. Now from the graph we can right FX0 is equals two five. F x one. That means here we have one F X two and it will be equal to one and FX and -1. Or we can write FX three is equal to two. Now substitute all these values in the formula of left remain some. Then we can write left. Riemann Some is equals two. Fx zero. That means five multiplied by it, two plus one, multiplied by two plus one, multiplied by two plus two, multiplied by two. So when we further calculate this, finally, we get left to demand some musicals to 18. So this is the final answer for this problem. I hope you understand the solution. Thank you.

Okay, Theo, first thing we want to do is find the length of each sub interval. So to find each sub interval, we want to find Delta X Delta of axes denoted in this way. And you want the top of the fraction to be the difference between the in points. So we'll take to minus zero over the number of some minerals that we're looking for, which is four. So we get 1/2 for our delta of X. Now, once we have our delta of X, we want to find the area under the curve. So, looking at our traditional general equation defined area of a curve f of X one times Delta X and so on, we can factor out Delta of X right here. And then we can go to the next step where we plug in 1/2 as our delta of X. And then we plug in our f of X, each equation for each of our different amounts that we're using. So for this 1st 1 here, we're plugging in 1/2. Therefore what? That is why it's pi over two and then plus fx of two, which is 1/2 plus sine squared of just pie because we're plugging in one times by and then the third, uh, part is 1/2 plus sine squared three pi over two because we are plugging in three halfs and the same goes for the last part. And so then once we plug that in, plug those values in and calculate them, we find out that each part is just 1/2 plus 1/2 which adds up to four in the brackets, and then we still have 1/2 out front for a delta of X. So multiplying those together, we get the area to be to Now we want to look for the average value of the of the sun. So to find that we need to put take one over the difference of the endpoints again so to minus zero times the area. And when we plug in our area to we get 1/2 times two, which gives us our solution of one


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