find-the-sum-of-the-areas-of-10-circumscribed-rectangles-for-each-curve-and-show-that-the-exact-ar-2

Question

Find the sum of the areas of 10 circumscribed rectangles for each curve and show that the exact area (as shown in Exercises $15-18$ ) is between the sum of the areas of the circumscribed rectangles and the inscribed rectangles las found in Exercises $5(b)-8(b)$ ). Also, note that the mean of the two sums is close to the exact value.
$y=2 x+1$ between $x=0$ and $x=2$ [compare with Exercises 6(b) and 16]. Why is the mean of the sums of the inscribed rectangles and circumscribed rectangles equal to the exact value?

Haggai Liu · Accepted Answer

Okay. So, uh, how to approach this problem? Let&#x27;s start by drawing a curve here. Ah, so say the curve will look something like that. Ah, and maybe I&#x27;ll dry twice. I&#x27;ll drive. I&#x27;ll draw the same cove again. Ah, so try to make it the same on Duh. So one of these, we were estimate using an inscribed rectangle. Remember? We&#x27;re trying to estimate the area inside the coat on one of these will estimate using a circumscribe rectangle. Okay. Ah. As you can see, when we estimate using an inscribed rectangle we have Ah, we haven&#x27;t underestimate on the stuff I shaded in black basically will be. The year will be an area. That Majesty error on here would have an underestimate. Okay, Um, so this is an open. We have an overestimate with this. Ah, with an error that&#x27;s shaded in black over here, on the right on that. You can see one of these is in overestimate. At one of these is an underestimate. Ah, And if we take the mean of these two rectangular areas, we will we will cancel out some of the overestimate with the underestimate. And hence, this is why we shall expect to get a more accurate estimation of the true area and they need the code. Thanks for watching

Matt Bremer · Answer

right in question number 18 were given. The function of the vaccine was X squared, plus two. We&#x27;re gonna try to find the area under that. Her using approximation of rectangles over the interval, 0 to 6, um six. Invented by that in six sub intervals. That means each of our sub intervals is each gonna be one you that preach some Venerable. Right. So, um, basically, what we&#x27;re gonna do then, is, uh, first talk about what&#x27;s the difference between our Ram and El Ram. So armies that you&#x27;re going to use the right side of the interval in order to be the top of your rectangle. And this means you can use the left side of the interval to be, um, the top of your rectangle. So, for instance, what we&#x27;re talking about here, iss Um, so far, first interval. We&#x27;re talking about this terrible years, So that one we&#x27;re going to use the right side, so our first box could be like this. All right. And so the base this is one times the height, which is gonna be f one. Just three. Okay, next is were too. Use the right side. So this is box muse. for that. Okay, so we&#x27;re gonna add to that the based one The height, iss six. Plus, you stop using parentheses for these. Um, but three, uh, this bus here. All right. And so that&#x27;s gonna be one times 11 plus and then the next one for we&#x27;re doing this. You noticed that I&#x27;m gonna get an overestimation for, um, one times 18 overestimation of the actual area under consolidate rectangles are little. These rectangles air little bit over the actual value. Thumbs. The next interval is by a year. That&#x27;s gonna be again. All these bases or one that&#x27;s that&#x27;s the width of our intervals. This will be 27 and the last one. Bye. It&#x27;s gonna be 38. Okay, so lines one times 38. So, Max, we have to add up all those values. C 36 plus 11 plus 18. US 27 plus 38 is 103. All right, so this the right side some. Okay, um, and again, that&#x27;s gonna be too big right now when we do the left hand left here. All right. All right. Um, one our, um So what I need is I need another picture of this good. Okay. Wow, this is what I was doing. All right, So now for the left hand, we&#x27;re gonna use the left hand side of the interval for these ones. So? So I did it. So my 1st 1 from zero, the one I&#x27;ve been used zero side. It&#x27;s gonna be this box here, so that&#x27;s gonna be one times two. And then the next one, I will use this side here. This is underestimating. It&#x27;s a big one. Times three. Then the next one here one time, six and next, A one times 11. Just doing, like times with breach of leads. And then one time, you, too. And then one times 27. So this is underestimating because I&#x27;m leaving out these areas right here. So there should be less than the previous one. So we get two plus three plus six plus 11 plus 18 plus 27. Randall was 67. All right, so then the final thing I want to do is get the average of the left and the right approximations. That&#x27;s gonna be 103 103 plus you seven divided by two. That&#x27;s 1 70 over, too, which is equal to a t. Uh, all right,

Ankit Gupta · Answer

the interval given to us is minus went toe to and we have to take one unit by rectangle, so the next in points will be 10 and one. So we have opted three rectangles off one unit. But function given to us is G X is equal to minus X squared plus eight. So the value off G AK minus minus, minus minus one whole square Class eight, which is eight minus 17 g off zero, is equal to eight g one will be could did you of minus one G off you will be equal to minus blue square less eight which is minus four plus eight which is equal to food. So let even be the area. When we take inscribed rectangles so even will be equal Do one into fo zero ls one in to f off one less one into air force do since she hit Does the function is G So therefore money to G 00 is running toe ate less one in energy of oneness seven and g off do is it will look for so this is eight plus seven plus food, which is 19 square units eight to be the area when we take circumscribed rectangles, so the area it will be one into G O minus one less one into G 00 plus one and two g o one g of minus one is seven. Gov. Siegel is eight and GOP one is equipped seven. So their food. This will be 14 less eight, which is equal to 22 square units.

Ankit Gupta · Answer

here. We have given that f X equals two x a square now, ex belongs to here. Available also zero four. No, we have to you first, it submission off and equals to one toe it. 0.5 unit and funds on off in. And then again, be somebody shut off and equals to one four one. Now we have to can clinical care cats there too. Estimate is the clue that do the actual area of the girl. So when we estimate and find here that yeah, it more grow that course and is much closer. The reason is very simple. If we take more number of the reason more number off division, then we will reach the more fine Would you tell Final pollution and we get final answer and more equity she

Ankit Gupta · Answer

No. Now we have to grab the car. Why equals two Wonder where with three x cubed. So, no, this graph can be then. Here. No, here were pressed. Why equals to one divide with three X Q and no, this is a coupons. Now we have to find This is this is the week off when you&#x27;re it. And this is the why you exist here. And this is the word excesses. Now we have to find the idea here. So for this, we have to find a delight inscribe direct in this matter. So these are the three rectangles. So one divided with three multiplied with one unit plus into the world with three multiplied with one and nine multiplied with one. So by solving, we get Wonder Bread with three it divided with three and nine. So this is here. So No, this is the area 12 Unity Square. And now this weird sort allocation

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Find the sum of the areas of 10 circumscribed rec…

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

Kayleah T.

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Samuel H.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus