use-a-graphing-utility-where-helpful-to-find-the-area-of-the-region-enclosed-by-the-curves-ysin-x-yc

Question

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=\sin x, y=\cos x, x=0, x=2 \pi
$$

Gregory Higby · Accepted Answer

Okay, Um So when heading, use a graphing calculator to show you, Um, the sign curve is the red one. That co sign curve is a black one. The vertical line x equal zero in the vertical lines. X equals two pi eso as you&#x27;re looking at this, what you have to be careful with if you&#x27;re not using a graphing calculator, is you want to make sure that you do co sign minus sign from zero to pi over four and then switch over Teoh do sign minus co sign from pirate for 25 pi over four and then switch back to co sign a minus sign from five power for 22 pi because that&#x27;s where they want you to stop. Otherwise, if you&#x27;re using a calculator, you can just, you know, find your function that has the in a role from zero to too high the way. This is where you find the high value in this calculator and what you want to do then is so you only have to do. One function is if you just do the absolute value, um, and makes everything positive. Um, now, the downside to this calculators, it does not give you an exact answer. Um, but if you checks with an exact answer and some calculus actually might let you get an exact answer, I have not seen this capability. It is exactly four route to, um But if you actually went through the process and you did zero to pi over four pi over 4 to 55 R four, then 5/4 to 2 pi and worked it allowed by hand finding values on the unit circle, you&#x27;ll get exactly that. Makes perfect sense for root twos. Your answer, by the way.

Bailey Brooks · Answer

for this problem were given to functions y equals co Sign X, shown here in blue and y equals Sine X, shown here in green and were asked to sketch the region between the curbs and X equals zero and X equals pi of to So from X equals zero X equals pi of two. They&#x27;re two different regions in like for area between the curves. For the first region, it&#x27;s from X equals zero. So this point right here the second reason region is from this point, it&#x27;s a pi over two. So to find this intersection point so we create these can create these two separate enter girls to estimate the area, we set the two functions equal to each other to find the intersection point. So we set co sign next equal to sign two x and find the X equals pi over sex and pi over do as well as further multiples and derivatives of it. So therefore, the first integral to estimate the area we from zero toe high over six of the function on top so co sign X minus the function on the bottom signed two X and then plus the next year of it, which will be from PI over +62 pi over two. The function on top signed two x minus The function on the bottom Co Cenex that&#x27;s written out here The integral of co sign X is sine X and the derivative I mean the the integral of sine two x is negative. 1/2 co signed two x, which to figure that out you can kind of work backwards by thinking, Well, what&#x27;s the derivative of like co sign two x and working that way to figure that out? Then the integral of sine two x is negative. 1/2 coastline two x in the integral of co sign. Next is Cenex. So we plug in pi over six for X from the first in a girl and the sign sign Pi over six is 1/2 plus. Then it would be co sign of two times pi over six, a co sign of pi over three, which is 1/2 then 1/2 times 1/2 is 1/4 minus parentheses and then we plug zero in for X. So the sign of zero is zero and then the coastline of zero is one time for 1/2 is 1/2. Then we add the next integral to it. So you plug in pi over two for X for the second. Integral. So the co sign of two times pi over two is just the co sign of pie, which is negative one times negative, 1/2 assed positive, 1/2 minus the sign of pi over two, which is one so minus one minus parentheses that we plug pi over six in for X. So be co sign of two pi over six or co sign of pi over three, which is 1/2 times negative. 1/2 this negative 1/4 minus and then sign of pi over six, which is 1/2 adding. All this together we get the area to be 1/2.

Sky Li · Answer

Okay. This question. We need to sketch the region closed by those two. Took to curves in a fine is area. So here is some x y plain, uh, And as we see the domains from zero to two pi or better focus On the right side, there are two pipes. The first size ko. Thanks. Because I actually look like that, huh? They know, uh, ex yuko&#x27;s to pie or two Cossacks equal to zero on the ax equals zero Kozak. Seacoast one. Here&#x27;s one goes like this. You&#x27;re a right one. This is X equals pi. It goes to why, Yeah, OK, but this will be all the co Xanax. And to Linus go sonics. Um, Linus goes on X and X uh, close to zero. This ecosystem one okay of the ex coast to pie. This actually counted three. So Jim uses three. Hey, it&#x27;s like they flip this costar X life is why you coast war and we&#x27;LL get our two minus co psychics So this functional look like tears. Hey, the rich to three. The ax he caused by then it goes down to what? Okay, then we need to find area in between. Okay? So we see all the things represented. Backpacks. We determine into use X as our our integration factor. So we take them to grow with respect acts I can have. You know, X goes from zero to two. By here is the thing inside Integral is upper Curve is minus the lower curb. So is two minus Cozaar Max minus Cossacks. There will be two minus to Cosette and Auntie do give office will be on to X miners. Two sidekicks. You added it at access codes to to pi. And he goes to zero over X equals to privacy close to four pi. Um, signed two point zero. So this term skull. So we got four pi and minus X equals zero. We know everything. There is zero. So our answer all before pie.

Vinnu M · Answer

So in this question, we need to find the area off wife Equal toe finding work X. Since we know that area off a graph, it&#x27;s the integration off. Why? With respect Toe DX. So this gives us indeed, Eagle off signing worth X DX. Now, to find the limits, we already know that x zero. So this would be one off the limits. And we also know that wife equal toe bye bye. Not since why is able to buy by two. We can easily say that X would be one. So the limits would be from 0 to 1 signing worse off X dx for no. After solving this, we have integration off signing with X as X fine in worth X minus square root off one minus x squared. But the limit course from with zero one so the area would be equal toe sign in worth. Well, the one and minus so glad wrote off and it&#x27;s off minus squared. Broke off one miles the little squared. But you would give me the area as bye bye to plus one square unit. Ask the area

Gregory Higby · Answer

Okay, So, um, based off of the directions here where we have co sign of two X, the amplitude doesn&#x27;t actually change. Oh, wait. That sign. Okay, so this is co sign. Um, but the fact that we have two X right here that&#x27;s saying that they&#x27;re squeezing the period down. So instead of being one, um, cycle being two pi one cycles, actually, pie. Um, So what are we looking at? Is ah, from pi over forward. A pirate, too. Like, right over here. So Right. This value is PIRA, too. Um, And where&#x27;s the other value? Pi over four. Okay, is half of pirate too. So it&#x27;s right here and Weichel zeros. This area is all we&#x27;re looking for. Um, now, if I were doing this, OK, so that piper for has your lower bound your powers tatoos your upper bound and it zero as your upper function like a zero. That&#x27;s Ah, X axis, in case you didn&#x27;t know minus co sign of two x, the X. Um but so you might be saying there and say, What&#x27;s the point of writing? Zero is you actually don&#x27;t need to write that up, But that&#x27;s why it&#x27;s a negative there. Eso if you do the Andrew toe, the some people would do you substitution. I wouldn&#x27;t Just because I know the anti derivative of negative co sign is negative sign. But then I would think about this is really the chain rule, cause if you did the derivative of neck of sign, you would get negative coastline, but then you&#x27;d have to take the drift of a to of two X, which is too. So you can see that I&#x27;m missing that. So you can fix that by putting one half in front. And that&#x27;s from power for too high or too. So then from here, you can actually plugged that in, uh, negative one half sign of two times five or two. Okay, So things were cancelled then minus negative. Um, one half sign of two pi or four. You see that? That would simplify to be pirate to s o. What a sign of each of these will sign of pies. The Y coordinate, which is the ordered pair. Zero went to pie, by the way. So this piece is zero. So really, What? We&#x27;re left with his name. One Halftime 00 Plus this value for sign of pi over two. We want the Y Coordinate is once of this piece is 11 half times one is one half. That&#x27;s your answer. What half?

Apoorva Shah · Answer

Okay, So the first part of this problem except asked us to sketch the region enclosed by these given curves. So we&#x27;LL start with that. Okay, Here&#x27;s where access will look like we&#x27;LL start with co sign of X and this clear familiar with So just mark down some key points we&#x27;LL start with I ever hear And then high over too Okay, so then you know Cem values of co sign for example A sign of zero zero equals one co sign should know either. Well, that sign of pirate too equals zero And finally, Mia Cosa Hai, and that equals negative one. So we&#x27;re gonna use these values Toss of help us Dar graph for the other function as you&#x27;LL see. So first we have these values So we can What? Grass? Yeah, And this and we know the general shape of the coast line graph. So look something like that. Okay, Now we want to draw our graph of one minus co sign of X. And since we&#x27;ve already figured out these values, we can already fill in most of it. So for one liners, co sign of X, we have zero. Yes, at high over too that that would be one minus pirated one minus one. Because the next one that is this and finally at high. Shit. Hi. We have that. It&#x27;s one minus negative one. So that is two. And I specify that. Negative there. This is too SNL. It&#x27;s villains graph. And again, Think about what this graph should look like. Well, it&#x27;s the coastline graph, but flipped. And then it shifted up one. So this general shape will still be that of a co sign graft. They look something like Okay, I just want to emphasize the curves a little bit more, So I will redraw this. It&#x27;s lovely. Okay, so now that we have the graph sketch, we want to find the area of this region over here in this region. Okay, so this means we have to first figure out what this point right here is because you have to integrate from here to here. And then we need Teo switcher graphs and integrate from here here between the first calculate where these two graphs intersect. So let&#x27;s do that. We have this gives us partner equals money was too close an axe or what has if coz next they remember that X goes from zero to pie. So the only solution we can use are possible values from Europe. I So, since that&#x27;s the case Oops. So that&#x27;s the case. You know, the coastline of exes. Pirate three. They&#x27;re not good. We know access people privately. This makes costar Vex. He could want of it. So now we know generally where into role look like so from zero to pie. Over. Three have co signs on top and one minus. Co sign is bottom surgery. Subtract. You know, bigness attract, um, one minus co centerbacks from co signing checks. Then from high over three to pie, we&#x27;re going to subtract coastline of X from one mind to co sign cheques. So on the next page. All right, the integral. So we have now. It is fixed in front. He said it looks cleaner. I&#x27;LL try squeezing the DX over here. No, not that, actually. All right. Right now it&#x27;s just a matter of simplifying the integral ls that we have? You are. You could I&#x27;m sorry. She relies on us to the next my sight. Genesis high over three one, right. She cooks and right there and then integrating this gives us and into a girl. Of course. I kn ends. She sign. Okay, we&#x27;re almost there. So remember, pirate So evaluating Sinan Pirate three. It&#x27;s just Let&#x27;s just write a some common values of sign. So weaken wisdom our calculations. So we have actually have never. It&#x27;s right here. So side note signs, however, three equals mystery over too. And sign pi there was one. Okay, So now what do we have? We have minus three and then plus two signed pirate Three, three, three. Hey, now we value is zero and I forget to mention what signers ears. But they just couldn&#x27;t do that. So we get minus zero for mind sex here. Maya&#x27;s Linus, you&#x27;re technically didn&#x27;t matter. And then minus two sign of zero zero. So this part is irrelevant. So it was just giving them. Okay. Okay. Now back to the first Interpol we have for the ten second grocer. You have what? Hi! Minus two signed pie to sign. Pi is minus two plus pirate three. Maya&#x27;s to sign Piratey. So that b minus reading. Okay, so putting this all together, we see that something&#x27;s cancel out. They have this cancels out of this. I can&#x27;t serve this so we&#x27;re left with. Our final answer is pilots too

Problem

Use a graphing utility, where helpful, to find th…

Problem 21 Easy Difficulty

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=\sin x, y=\cos x, x=0, x=2 \pi
$$

Answer

$$
4 \sqrt{2}
$$

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$$4 \sqrt{2}$$

Calculus Early Transcendentals

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Catherine R.

Caleb E.

Kristen K.

Joseph L.

$$
4 \sqrt{2}
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals