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7. [4 points] Discrete integration Thc area under Cuve f(r) betweenandcan be approximated by the mnid-point ruleS =2'(+46-H))a , with 4 = (6 a)N_ For a = and 6...

Question

7. [4 points] Discrete integration Thc area under Cuve f(r) betweenandcan be approximated by the mnid-point ruleS =2'(+46-H))a , with 4 = (6 a)N_ For a = and 6 = 2 and the function f (r) = 12 21" , compute S for N = and N = Compare these results to [ = f f()di:

7. [4 points] Discrete integration Thc area under Cuve f(r) between and can be approximated by the mnid-point rule S = 2'(+46-H))a , with 4 = (6 a)N_ For a = and 6 = 2 and the function f (r) = 12 21" , compute S for N = and N = Compare these results to [ = f f()di:



Answers

In Exercises $23-26,$ use the Midpoint Rule with $n=4$ to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=\frac{1}{4} y, \quad[2,4] $$

And use the big point roll on this function there. This interval longtime many right angles to find out what tell taxes just going to be for four minutes. 0/4 she calls one. That means normally it would be from 0 to 1 to 2 to 3 to four. Since raising midpoint role we want one half, three house, five house and seven half. So then from here Do one times half of one half Plus f of three amps Plus half of five house I say from seven house. So that's going to be equal to 11. Plug that into the original air. Let's go ahead and find out what the actual integral this is. So from 0 to 4 for why I asked why squared That's going to be able to forli squared over two. That's why I cubed over three. Taking that from 0 to 4. So that's going to be equivalent to 64/2, 64/3. That's equal to 32 or three just equivalent to 10.6 heating. And we can compare the two And we don't definitely know that 10.6 Painting is less than 11. That's the comparison there.

The given question, We have to find the idea off the region. We can find it from the integration off effects, the X from zero to wait. So here the value off a street will you visit and the value of any signals to it. So the sub interval is one right. So with the help of trapezoidal method, we have relation. I hate ministry upon to into it after you know, plus 12 1 plus to have toe plus to have three class upto 12 7 plus have feet. We can read the values from the graph and we can subsidy substitute. We'll use We get jail plus two into one point. The plus 2 +123 doing to 5.7 going to get 0.9 plus two into them, plus two into nine, plus 26 plus trader. When the simplified is you get the result. As for 3.9 now we can find the with Delta Simpson method. So we have the exhibition for a definite in trickle. That is, the administrator appointed a into it after you know, plus four last one plus to have to plus four left Lee plus two of four plus upto F eight right so you can substitute the values from the Gulf. We get the final result as 44.6. So there yes, from the two method the episode and Simpson, a 43 point nine and 44.6 respectable.

In question. We have to find the idea off the idea after share that region. So with the help of representing rule, the expression for the area is jail toe for the effects. The X This is almost ableto four minus trailer upon going to fall FX, not plus defects one. So I do f x one plus two effects, too. Plus two FX three plus FX for Yeah, that's not a psychosis ego X one is equals to one X to it because it's so and so on. We can lead the values off the we can lead the values off the from the graph, and we can plug them into the equation. So we can we get these? Yes, one by two three plus to go into seven plus two into nine. Last two into seven plus judo, which sickles to 24.5. So this is the idea off the region with the help of trapezoidal method. Now we'll find the with the help of Simpson matter. So the relation the expression is four minus studio upon three into food. Uh huh, FX not plus four effects. One plus to wear effects, too, plus four defects tree plus effects, for we can read the release from the graph, we get one by three and two three plus four into seven, plus two into nine, plus four into seven, plus judo like, which is almost equal to 25 point 67 So the areas some trapezoidal method is 24.5 and some Simpson method is 25.67 okay?

So for this one is supposed to use the midpoint rule on that interval time. Any rectangles? It's fine till to experts until tex is equal to four minus 0/4 chuckles one. Just nice because when we do 01234 point rule wants us to find the middle of these. So then you have one half, three halves, five house and seven house. That means we have one times half of one half plus half a three house plus half of five house plus half of seven house. Okay, that's equal to 25 when you plug that all in there. So you're playing out to the original function here. It's going to fight with the actual Angela all of this. This is the integral from 0 to 4 of y squared plus one. Dy, just according to y cubed over three plus y taken from 0 to 4. That means we have 64/3 plus plus four. That's going to be equal to 76/3. which is equivalent to 25.3 repeating. So we could see that 25.3 painting is greater than 25. That's the comparison. The answer to this one. It's the answer to the midpoint rule here and return.


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