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Question Find tbe area of the shaded region A Fo) _ 9)_ dx Jzsi" - 3x)Jx'2Siw 3x-1 ) dx38 2 sin2Sjm 3x-T)82 sin 3 (2) 7...

Question

Question Find tbe area of the shaded region A Fo) _ 9)_ dx Jzsi" - 3x)Jx'2Siw 3x-1 ) dx38 2 sin2Sjm 3x-T)82 sin 3 (2) 7

Question Find tbe area of the shaded region A Fo) _ 9)_ dx Jzsi" - 3x)Jx' 2Siw 3x-1 ) dx 38 2 sin 2Sjm 3x-T) 8 2 sin 3 (2) 7



Answers

Find the area of the shaded region. $r=2 \sin (3 \theta)$

In this question. Her figure is given to us in the caution we have to find the area of the reasons are bounded by the crafts of why sign your bags and why it was caution cube bags in the in the world on the in the world -3554. By by four. No area of the reason is given by the integral Hussein cube X minus. Thank you. Becks The X Limit -3554. Bye bye. For now we will apply linear T here so we can write the integral esque designed cube X. Dx minus integral off. Thank you. X. B. X. This global market equation number one first Vivian stoll. Thank you X. B. X. This we can graduate as integral 1- School Science Square X. Multiplied by cynics B. S. Okay all we have use the biometric identity Science square X plus cosign square X equals to one. Now let us assume that U equals two Jose. Next on differentiating it. We get to U equals two minus find next E. X. From from here bigger B X equals two minus one. Divided by Cynics. Ew No they will substitute this value and are integral becomes negative one minus U squared ew. No. In order to remove this negative we can write the integral s integral. Use where do you minus integral do you On integrating it? You get U cubed divided by three minus you. Best constant city. Now we will substitute the value you here so it becomes assigned you X divided by three minus cosine X plus constant city. Now we will solve integral. Co sign your bags bx this weekend. Deputize integral, cozy. Next multiplied by one minus sine X. Science square X B X. No Let us assume u equals two finex on differentiating it bigger deal it was to person X E X. Now we will substitute these values so are integral becomes one minus U squared you in spirit form we can write to desk integral to you minus integral of U square bu On integrating it we get U minus you. You Divided by three plus. Uh huh. Only substituting the value of you. We get cynics minus find a few bags Divided by three. Let's concentrate to see Now from equation # one began diet identical frozen cube x minus find cube bags. The x equals two integral. Wasn't your bags B x minus integral. Sign you bags E X. We will put the values here by next minus. Thank you back divided by three minus course. Thank you Eggs divided by three plus. Co sign next first constant to see now we will apply the limits so we can write integral for Science. You bags minus sign cube bags. The X limit minus. Do you buy that for? To buy a rifle equals two. Find five x 4 minus. Thank you By therefore divided by three minus cosine cubed. Bye bye. Full invited by three plus was signed by by four minus sign -3 by by four minus. Thank you -3 by by four divided by three minus cosine, cubed -3 by by four, divided by E Plus for Science -3. Bye bye four on simplifying it, we get around so which is five times under two divided by three. Thank you.

All right. Number 96. Alright. With you is you go toe to X. Do you two d x and the new limits five or six. However, to then we set a brand a Groll. Okay, then evaluating this becomes one half. I'm CSC you. I waited from five or six. However, to that comes out to one half times negative one plus two or one half.

Ah, good day, ladies and gentlemen. Um, today we're looking at Ah, problem number 72 from section 4.5. And it's another. It's another definite integral that we wish to evaluate. And, um and I think, Ah, the message I'm gonna that I'm gonna use is probably the best one to use at least as far as I can think of. So let's dig right in. Uh, uh, Note that in this case, it's really a two step in a girl. So on the first step, what we're gonna do or what I like to do is deal with the two acts. So I'm gonna turn this I'm gonna use you equals to two acts. And then I replaced. I changed the bounds of the integral note. Um, in case you can't quite tell this, the top is pi over four. And the bottom is pi over to 12. No, it's ah, little hard to read, but but that's what it is. You know, eso um you a pi over four, of course. Is pi over two. And you a pie over. Well, this private six and ah, Then I get 1/2 to you. Is the axe of course. And now the next step is I'm just gonna plug these guys in, and I get the new integral here, Um, which is just, ah, pi over six on the denominator here. And, um, pie over two loops and see if I can do it here. I Yeah. Sorry about that pie. Over to the They're okay. And next, then, Ah, I'm just rewriting this, uh, again. Sorry. If you can't play well here, I'll move it down. Um, moving down here. Ah, but ah, so yeah. There we go. Okay. Yeah, it's ah, by, um, it's signed. Now I've just rewriting. This is sign and co signs because Cosi Kent is one over sign. So looks, uh, CO c can't, um, coach, she can't is one over sign. Of course. So close. He can't of you is, um, sign you to the negative one. And, um, times on, then. Ah, boy. Sorry about this. Um and, um co tangent coat hands and is ah, caught. I guess you is co signed. You, uh, signed Negative. So when I multiply that out, of course, um, I get it. I get this, I get this right here. So, um I get signed negatives signed you to the negative second coastline. You and 1/2. Um And now, of course, if you you can look at this and see sort of what the next step is, Um, sort of Just by looking at it, because the co sign, of course, here to the power and the are a Sorry. The sign here to the power and the co sign here. Coast on you. Do you, um, pretty much tells you then that you can make a, um, substitution. Just using JIA calls to sign you. Course. Um, yeah. So and that's what exactly what we're gonna do. Um, calculate this. I just subject to directly geek will sign you. Um and ah d g is coast on you. And of course, the bounds are nice. There. One looks, uh, no wants to do that s o you get 1/2 and one. Um, and now with all that done now, all we have to do is set it up. Um, and, uh, going here from 1/2. Ah, to one, of course. Ah, and now, really, all you have to do is directly apply the, um, generalized power rule for entered rolls, and that's all you have to do. Uh, so all you have to do here is I'm just a close. Okay? Come on. Come on. Okay. Yeah. There we go. Um, yeah, it's all I'm really doing is applying directly the thea which McCaughey Um the rule, huh? The generalized power rule here. So it's from 1 1/2 Um, and then Really? Oh, sure. Um and then Okay, once we have that, now we just evaluate. We just evaluated at the ends and we get, um you're going to get 1/2. So So, yeah, I guess to summarize, basically, it's gonna have a two step integral. There's really two substitution. Tze the 1st 1 that I do or like to do is sort of get this integral to this form here on. And then once I get it, they're the next step. I just handled the co signs and signs here, so that's kind of the way I do it. Of course it's not required. You do it that way. You can sort of do it your own way. Um, whatever is most convenient to you, but for the sake of demonstration, I prefer to do it this way. Uh, anyhow, that is it for this problem. Thank you very much.

Consider the graph of the region below. Now to find the area of this region, we first have to decide which strip should we be using. So if we use, okay, lets say vertical strip, what happens is that this region will be divided into two. Because if I draw a vertical strip over here and I draw a vertical strip over here. Looking at these trips, they will be using different upper functions. So the first one will be using why equal sex. But the second strip will be using y equals one as an upper function. So that will be lot complicated than when we're going to use a horizontal strip. Because when we use a horizontal strip, say this trip over here, all throughout this region, the strip will be using the same right and left function. So this will be easier because you will only have one integral to define the area of the region. So our area A will be equal to the integral from A to B. All the height of this trip times the width of the strip. Which in this case will be in terms of why? So we have a Dy And since our and the rule is in terms of why then the A and B will also be Values of Y. Which in this case will be zero and 1. So we have integral from 0-1. The height of the strip will be determined by the difference between the right function and the last function. So since our functions are why in terms of X, we had to rewrite that into X. In terms of why? So the first function here, Y equals X. That's the same as X equal to Y. And the Y equals X squared over four. We can read that into four, Y equals X squared. Or if you take the square it we will get X equal to The Square Root of four. Y. You will use the positive value of accents were in the first quadrant and this is equal to two Squared of why? So then the height of the strip will be Difference between two squares of Y and why? And then you have the Y. So now integrating me get to times square to y. Is Why raise to one half sole integrating that. We get why raise the 3/2 Over 3/2 minus we have. Why will be y squared over two. This will be evaluated From 0 to 1. And then simplifying this, we get for over three times a wire raised 3/2 minus Y squared over two. That's evaluated from 0 to 1 and plugging in one for why? We get 4/3 minus one half. And then why equal zero will just make these zero. So you will only have 4/3 minus one half, which is 8 -3/6. That will be 5/6. So this is our area for this region


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