Find the area of the shaded region. 3)y = cOs2 x1y =-COS X...

Find the area of the region.
$y=\frac{3 \cos x}{1+\sin ^{2} x}$

So here we have the interval from zero power for a sign. Explicit, ghostly next minus coast annex. So just sign next. That's the integral from power force. However, to of science, please co sign next my sign X Just close eye next. Yes, The first part will be negative. Co sign X I, however, 40 uh plus sine x Well, you have however, to and high before like is ice uh native skirt too over too. Plus one plus one minus. It's great. You over too is as to minus Ritu.

In discussion. We need to calculate the area uh of the region enclosed between these cubs. The cubs given are as vehicles to cosine X and X vehicles to one minus three exhaust pipe. And the third one is exit goes to privately which will be a straight line parallel to the axis. So first of all we will plot the graphs of these equations, so we plotted the graph uh for these three equations, this one is for what he calls to Cosine X with purple ca and it's uh straight line is for what he calls to one minus three over X. And this state line which is red is for Mexico stuck by battery and we all associated the region between these three cars. Now we can calculate the area by partitioning the X axis. The this region exchange from Mexico's 20 to access also by by three. So this area will be, can be calculated by integrating for the limit 02 uh by by three and uh as we can see the a pack of is here, this is cousin X. And the lower cup is the straight line which is why it calls to one minus three over PXE. So we can write it here as ffx minus the fx or dx. So we can write it as cosine X minus one minus trip over PXE will be minus one plus three over pipe, X dx. Now we can integrate this and we will get the area it calls to ah by integrating this, we will get cosign X will be integrated to Synnex, one will be integrated to X and Plus three over par tax will be integrated to three over pi access care by two. That will be 3/2 pi. Not access square For the limit 0 to private tree. Now by executing this limit we will get the area s uh area they will be called to when we will substitute their parliament. This will become science private resigned private tree is Route three x 2 minus X. Will be privately only and Plus three of them to buy excess care. This will become plus 3/2 Pi. Multiplied by pi. Betrayal sky will be pie scare overnight. So this area they will be called Here. Route 3/2. This will be -9 bigotry and this will become a plus. Sorry here we forgot to calculate for the that for the lower limit it will be zero so we ignored that here and here. This will become um three By over 18. That will be plus by over six. Now, by simplifying this, we will get the area is Route two x 2- by by six. So this is the area of the shaded region which is the region between the given cubs This isn't a square units square units. I hope all of you got discussion. Thank you

Hello. So here the area of the region between what we have Y. Is equal to three to the coastline of back center backs. We have wild 0 to 0 X. Is equal to zero and execute a pie. That's gonna be the the area here is the integral from zero to pi of three to the coastline of X. Times sine of back to the X. So to do this integral we're going to use a U. Substitution and let um you be equal to just co sign of acts. We have co sign of X be equal to you. And then we get that negative sine of X. The X. Is equal to D. You. So we get that when X. Is equal to zero U. Is equal to one and when X is equal to pi U. Is equal to negative one. So therefore we then get here while we get negative the integral going from one negative one of just three to the U. D. You. Okay so we're valuing this integral. This is going to give us um Well we could flip the basically instead of having negative integral this could be positively integrated. Instead of going from 12 negative one. Now we're going from negative 1 to 1 of three to the U. Do you. Okay so we evaluate that integral and we get just three to the U. Over the natural log um of three. Um And then we evaluate that from 12 negative one. So so evaluated from one Over negative 1-1. So that's going to give us three over the natural log of 3-1 over three times the national of three, which is going to give us eight Over three times the natural log of three.

Okay, let's try to sketch the region close by those two curves. And since the domains by a specified only, I need to call the curve inside That dont make so our export. Here's our x y plain. No, this is good. Okay, look said this It's over. X y plain or ex goes from negative power three to power through it. Okay. And so of X equals power three. Tension to X equals two squared three No, the ex ecos inactive power three tens Next because through back to square root. Right, Except with zero A zero and lee no tension axis at our function. So it looks like this. And for on the other hand, for the second her are two science, you know, sex look like this so we don't We need to find the end points so and points is over X equals to power three. We know science is screwed three over to have a book Piper too. So it will be square through it assed we can see it was to curves Army intersect Had those toe end point who saw a region because Tucson accidents also our function so it will play it right here here is already close the region. And as we mentioned, everything here is symmetric by this region. So we know those two parts has I have exactly same area. So we only need to calculate one off them. The more diaper too. So say I want to calculate this area. They will be integral with respect to X. Since those two curves can be represented by X and the up the boundary here is goes from zero to three power suite and things have been integral. Will be upper curve minus the lurker, which is to sort of fix Linus tension Tex okay. And instituted off this So just mine is to co sacks minus the art. Andi, do it for ten years. We know that anti do it. If attendant attention access Cave lock actually cause annex. So this hoop clause log Who's Alex? They value it at power of three two zero so well, actually cause power three. This is over half. There's just two times previous form plus log minus X equals material is it's two times conectiv to here because cause and zero is one And we know Logue y zero so minus two plus zero. This will return Here is a plus for minus two is two and a plus two times log over too. Bye. And no for this log, we can bring one bag to sign to make this love too. What I'm saying? Here we have this for a general log. It's like negative bog A over B. Did he close to lock Good or anything? All right. It's like that. You re bringing in the negative side. Take the reciprocal inside. So this were also he close to two minus two. Block two. Yeah, We're both too can be our answer.