sketch-the-region-enclosed-by-the-curves-and-find-its-area-x2y-xy-2

Question

Sketch the region enclosed by the curves and find its area.
$$
x^{2}=y, x=y-2
$$

Gregory Higby · Accepted Answer

uh, not sure where they wrote the problem this way. Um, but y equals X squared. Looks like this to probable. I think everybody knows that if you&#x27;re in, if you&#x27;re in countless, um, and then the other function X equals Y minus two. Um, I write as why equals X and you have to add the two over, Um, so, like, shift up to and the soap is up 1/1 line like this. Now, here&#x27;s the issue is we&#x27;re definitely find in the area between these two, but we don&#x27;t know those bounds like I don&#x27;t know this x value in this X value question mark Question Mark. So you find that by setting them equal to each other, X square is equal to X plus two expo area and subtract X. I&#x27;m gonna subtract two over. I can look at this as a quadratic that a factors A to, um one into that had to be native one. Well, that works of it&#x27;s plus one minus two. So I get X plus one next minus two, which means my bounds and this should make sense is negative one and positive two. So if you plugged these and you&#x27;ll get the same. Why values A one plus two is one. They won&#x27;t square this one. Two plus two is 42 squared is for okay, so don&#x27;t checks out. So those are my bounds from native wanted to. My upper function is the X plus to function and you need to subtract off the lower function, which is the X squared. And so you find area The X man From here, it&#x27;s just the anti derivative. So the anti derivative of X, you add one to your exponents and then you divide by that new exponents plus two x again, you had one to your exponents and divide by your new exponents. You can double check the derivative of this needs equal this so and it does You just believe me from negative 12 tubes. And now we need to plug in your bounds. So, you know, plug in to when you square it becomes four. Two times two is four. I&#x27;m plugging into, by the way to Cuba&#x27;s eight thirds and a minus. When you plug in negative one into all these negative one. Once you squares positive two times able in this minus two and then a one cubit is still negative. One so changed that to be close one third. Now, from here, I would go to a calculator cause you&#x27;re gonna get some fractions and just you can go with me as I&#x27;m this left wipe in parentheses here is 13th. Um, and when I get on this right from parentheses is negative. 76 And so going back to the 13th minus negative 76 you get an answer of nine house or 4.5. I like fractions.

Amrita Bhasin · Answer

first off to sketch the equation to X minus X squared. Give the integral from 0 to 2 of two x minus X squared Detox. Hello again. The minus zero Doesn&#x27;t matter. We end up with 4/3.

Gregory Higby · Answer

Okay, so, um, if you were to graph this, um, I believe it&#x27;s to over one minus, uh, one plus x squared, And then the absolute value function looks something like this. Yes, I&#x27;ve come put a rides on this. Um, so that&#x27;s the absolute value of X s. Oh, what I would do because these air even functions. I hope that makes sense to use. I would just find this in a girl and then double your answer. Um, so how do I find that integral is? Ah, it&#x27;s the general from zero. Because that&#x27;s C X and ah, were these two intersect? Um, is that X equals one? Because if you pull you one for X squared is one plus one is two to divide by two is one in the absolute value of one is still want eso The upper function is that to over one plus x squared minus. This value is just, you know, x at this point, um, because of ah, you know, we dropped the absolute value because we&#x27;re only looking at positive values for X. And so then when I need to do as I mentioned earlier, is double that answer. So Now we can go ahead and take the enrolled that so That too, is still in front. So the I hope you&#x27;ve already learned this otherwise it would take me forever to teach that that anti derivative is the inverse tangent of X. You can just google. What is the integral of 1/1 plus x squared? If you didn&#x27;t know that now this too is just a constant, so we can bring it in front. And then this in a girl, you add one to your exponents and then you divide by that new exponents from 0 to 1. So what do we have is now we need to plug in one and for all that. Um So where on the unit circle? Um and then to inverse tangent of zero. I don&#x27;t actually write in zero for this one. Actually not right for that one either. Zero squared is still zero. So I ran out of room. But where in the inner circle, this tangent equal one. Well, that&#x27;s where signing coastline are both the same. That&#x27;s that route to over to route to over to And that&#x27;s the radiant measure because that&#x27;s what this question saying in verse. Tangents. What&#x27;s the radium as your pi? Over four. So what we have is two times to hire or four minus one squared is still one times one half is one half now. Same question. Where is Inverse Tanweer&#x27;s Tanja equals zero. Well, that sign over co sign. And that&#x27;s the radium measure. Zero. So this happens to be zero as well, son. Just not gonna write it. So from here, just distribute that to enable two times two is four. Divided by four is canceled. So I just love to pie in two times. One half is one is our answer on this.

Sky Li · Answer

Okay, let&#x27;s get the region the first functions. Stand there. Two problems now. It just looked like this. Oh, pay up. So this is our Y cosa x square or a second function? Also a proble and we can&#x27;t find find. Ah. Then why you go to zero X Either ecosystem for or zero across the region and for zero. Okay, this is our xx, sis. This is our by axis, since since the coefficient of X square is an active, you know, open don&#x27;t downward. So there&#x27;s probably something like that. Okay, So this is our why you coast to for X minus X square and the enclose regions red there by those two problems. So we need to find out the Intersect, those two intersection points and they already know they both cross the region, So we got one. So what about the other one to get the other one? He said as, um this point satisfy both functions, so you know, x square ecos. Why, Nico? So for X minus X squared. So we know X square on this point. Thanks. Thanks to all right. Well, do you know this point as x two so x two square should be close to y two and buy two ico. So for eggs, too. Linus x two square. And we can&#x27;t get our AKs to from here. Our ex Who will be this? X two is not zero. We can cancel with blacks, too. For extra rico&#x27;s too. Okay, um now, let&#x27;s try to be better this area, since everything&#x27;s represented the ex, we decide to buy the integral with respond with respect to ACS. So this integral and bad. And we got this DX and for bond. Dre, we just figure out it&#x27;s just it goes from zero to two, okay? And the thing inside the injured girl is the upper curve, minus the lower occur. So hyper term being always for X minus. Expert, this is our op. Occur minus the lower curve. Which exact square. Mmm. So this then we need to you venerate this definite into girl. I&#x27;m too. I&#x27;d better with this. Definitely in the world. Um, it&#x27;s just to find auntie duty of face. Um, and the beauty of this will be the first. Hermes just tow acts where? And this isthe minus two X, where if they combine those two together, so the ante Dios you with that ability to third execute and you can take devoted to check. So if I take my free take diverting with this term, this will become negative. Two three canceled checks. Where exactly? Match with here. So this is our auntie. Do it too. And then the benefit at the bone chri actually close to and X equals zero. We know Plaxico zero this thing Nico to zero. So we only need two. You&#x27;d better it at two. Next he goes to the first term is two times for, say, the second term here is to third times eight because to Cuba say this will be on eight over three.

Fuzail Shakir · Answer

All right. So for this question, we have X squared and negative X squared plus four x You have X squared is equal to Native X squared plus or acts we have to find the intersection. Points just becomes X squared plus two x minus four X is equal to zero plus x squared, so this is equal to two X squared minus four. X is equal to zero with a factor of two. X to get X minus. Two is equal to zero, so we have actual equal to two on X is equal to zero. So therefore we can rewrite the integral as the integral from 2 4 0 2 2 0 2 0 0 to 2 of the ex queer minus negative X squared plus four X is equal to be integral from 4 0 to 2 of x two X squared minus four x This is to third X cubed minus two X squared months. We evaluate this from zero to and that that this is equal to eight over three

Catherine Ross · Answer

sketch the region enclosed by the given curves and find its area. So we have two functions. Why equals X to the fourth and why equals two minus the absolute value of X on dhe, they are graft over here in green and blue. On the right hand side, when we want to find area and we&#x27;re trying to construct are integral To find the area that&#x27;s enclosed between these two graphs, we need to also consider our boundary points. So we need to look for those points of intersection where the two graphs intersect and from the X Y tables we can see that at X equals negative one y equals one and X equals one y equals one. We have points of intersections where these two graphs indeed intersect each other. Now what&#x27;s so beautiful about these graphs is that they are symmetric. And what&#x27;s interesting about the blue graph is that this is in fact, a piece wise function. The absolute value of X is so if we split the blue graft down in half by the Y axis on the left hand side, we get the function F of X equals X plus two and on the right hand side. We get the function G of X equals negative X plus two. And this is helpful because since there is a cemetery involved in these graphs, there&#x27;s also cemetery involved in the region enclosed by these graphs. So, in fact, we can just calculate the region of half, and that region will call it our, since he&#x27;s on the right hand side is the same area as the region on the left. So all I need are the boundaries for the right hand side, and I can create an integral from that, multiply it by two, and I&#x27;ll get the area for the entire region. So the boundary, the lower bound for the right region, would be X zero and then and that&#x27;s because of the Y axis. That&#x27;s where we&#x27;re starting. And then for the right hand side, the upper bound of the exes one. And that&#x27;s because of the point of intersection there. That&#x27;s the X value. So in constructing the integral, we would have to if my panel work there, we go to times Thean, a girl from the lower bound zero to the upper bound one, and then we went to subtract the lower function. So the green function from the upper function from the blue function. So we want G of X minus hour axe to the fourth so g of X is negative X plus two and subtract X to the fourth from that. And that&#x27;s where the respect to X so d of X dx and then we have two times negative 1/2 x squared plus two x minus 1/5 x to the fifth, all evaluated from 0 to 1 That gives us two times. Now we plug in the upper bound first and then we subtract the value resulting from plugging in the lower bound. So two times negative 1/2 plus two. No, fix that. Here we go, minus 1/5 that minus plugging and zero. Well, that all goes to zero. So we have to times we want a common denominator here. So we have negative five over 10 plus 20 over 10 minus two over 10. That gives us two times 13 over 10 which two in 10 cancel to visit to once it wasn&#x27;t attend five times. So we have the area between those two curves. Is 13 a year five. That&#x27;s the total area between the blue curve and the green kerf

Problem

Sketch the region enclosed by the curves and find…

Problem 12 Medium Difficulty

Sketch the region enclosed by the curves and find its area.
$$
x^{2}=y, x=y-2
$$

Answer

$$
\frac{9}{2}
$$

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Michael J.

Integration Review - Intro

Definite vs Indefinite

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$$\frac{9}{2}$$

Calculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Michael J.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Michael J.

$$
\frac{9}{2}
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals