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Sketch the region enclosed by the qiven cunves peclde whether typical approxlmating rectangle Find the area the region, 6y2,Integrate wlth respectOr%. OrarStcp 1Ske...

Question

Sketch the region enclosed by the qiven cunves peclde whether typical approxlmating rectangle Find the area the region, 6y2,Integrate wlth respectOr%. OrarStcp 1Sketch the reglon;

Sketch the region enclosed by the qiven cunves peclde whether typical approxlmating rectangle Find the area the region, 6y2, Integrate wlth respect Or%. Orar Stcp 1 Sketch the reglon;



Answers

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ x = 1 - y^2 $ , $ x = y^2 - 1 $

No. Okay, so for this question, first of it all those curves Ah, so this is X y This's X axis and waxes and we draw X equals to one issues that vertical line here. And I keep Mexico's nike of one very sun in another line. And ah, for this proble X square minus one is here. And this exponential function into our axes here, right? And in the intersection here is because why you goes what on? So the shaded area will be here and ah, since everything given us represented about X, we will inter great made the respect to ex okay, interest x instead of why then we need to draw a typical approximating rectangle, Uh so probably re choose here Here is a small approximating rectangle If we scale, it probably looks something like this and the world will be out x And the height will be, um X the upper curve. Ah, riches you two the power X I miners on X lower curved excise square wireless. Okay, so this will be over typical approximating rectangle on after we read down this We know the integral is basic is based on this. So we know the shaded area of this region, eh? Now Kiko's Tio and grow ah, into the power X miners like square minus one. The ex The axe goes from negative One toe there. Now, this, uh, replying the anti dote for this that you will be in to the power X exponential. Function yourself on minus one over three X cube plus X and a battery. This one to an act of war since. Okay, so let's try this. So we know, eh? There he goes to the first one will give us no minus. Minus one third class war teaches the class to third Linus. Second one is plucking actually close to ninety war. So you two power ninety one plus here is plus one third minus one. So it's minus to third. So this will give us the finest one already. Class four, Spitter. Okay, so this will be our answer here

All right. So verse every draw X y excess and go the forest. First function, like owes one over X Terry's one. What? Okay, So this is why you go to while Rex and we know except goes to Ah, this vertical line here. And what, over x square? Um, so, yeah, you know, X square acts is bigger than one. It goes faster, ex Clarice. Bigger than acts. So the reciprocal of X square of a Latin will be less so she do something like this, All right, Because this point, um, or lower acts corresponding to where half at this point here. The access codes to of it. Plug in May, we called one fourth a quarter. Right? So that's why this curve is lower is that while we're act square and up her wise power x okay and the enclosed regions right there get, um, and everything is represented. That ax. So we decide. Integrate respect to X star treks on. Then we draw a typical approximating rectangle. So say the rectangle is right here. I know the zoom out. It will be something like this. This is approximate rectangle near the callousness Delta X and the height will be the upper curve minus the lower Kurt is one over X I minus one over X eyes. Where? Right. So, um or less Jarvis to find Ariel, this region and then by this formula we know area the egos through the into a roll. Um, with respect to X. So right now our jack's here, and then they know our X goes from one to two because the intersection here is one one, and, uh, they stopped at X supposed to. So it goes from one to two, and the integral inside it will be well over X Will be this without the subscript. Uh, well, our X minus one over x square. Then we need to find Auntie. You're okay about this? You know what? Or ex Ante Dios, you allow our access log X. So, Sophie, Log X and, uh, no Aunt Aunt Edie narrative for negative One X square. It's just while Rex. So this whole thing Nandi directive off this whole thing will be log X plus one over X. And I would take the boundary matter whose X equals two and actually close to one. So this is very close to love too, Because one or two minus love one, which is zero plus What right? This area. There will be love too. One half minus one is minus one. Yeah.

S o for this question. Less draw those curves. Oh, here's a X waxes And the first function Why Yuko's X minus two squared. So it's actually, while you go to X squared shifted to the right by too. Right. So here is to and the function go just look like x square. Yeah. Okay. And, uh, why you coach, your ex is a line A straight line. This is our blind ghost packs. And the first wise why you go to X square. Okay, so are you, huh? Our area are in close the regions here, right? So since everything is represented by X where oh, integrated with respect to X and ah to draw a tiptoe approximating rectangle So say we call a rectangle right here. And I was soon now the this Iraq Congo. So this will be our typical approximating rectangle there. It's a direct tango, right? So they both will be Delta X. Onda had the height. The head will be on the upper curve minus the Laura Kurt. So the average curries x I and our Laura curve here is acts I square So time, Linus Excise square. Then we can't write down our a formula for the area off this region area off this region A They rico's too well, um, And to grow with respect to x And we need to find the boundary for acts we can see from the picture. The bungalow A star from here x one here x two. Okay, so we know our interview over goes from X one two ex too. And I will figure out those x one and x to us later. And ah, a girl with Putin isjust this height here this x I minus x i square. So we put here. He's just acts minus x square. Okay, now let's try to find explore i x two So we know ex lax to are just of the intersection for those to curves. So we put them Tio So which means or ex I to satisfy both equation? Ah, you know, other worse. Ah, like I if we put into the Oh, sorry. I made a mistake here. This is X minus two to the square. So actually here is so here is X minus two squared. Right. So the height it will be x x I minus X minus is too squared and here change it to X minus two. Squared right, um, and to find those x one x two but plug into the equation. So we basically they got excited minus two squared eco's too X I, In other words, the typical approximate age teen rectangle at X one and X to the height will be zero. Right, as you can see that the shrink down to no height here is just a dot In fact, it's just a dot So there's no hide. In other words, this you go to zero. So this will give us this formula right there. Then we can solve for X one x two x wise, the smaller one extra. It's a bigger one. So let's do that. We take the square, this will be Max. I square minus four eggs I plus four. He goes to x I So if we move this ex side to the other side will get minus five x I they're, like, kind sold by factory. This will give us X I minus four times x I minus one Nico to zero So we can see from here our x one Yukos want and I x two seacoast on four. Okay, then we can change this X one. It's actually because one ex to ico c for and they saw this integral. We evaluate this into a girl, so we need to find Auntie derogative off this. Okay, let's do it in the next page. Our area A here. No Eco's tio and your girl. It goes from one to four. Thanks. Minus thanks. Minus two square the ex. Okay, so the anti dude here on this, the first Hermes just one half x square and second term here is actually minus one third. If the using the channel, we know that. And he do tell that, and they'LL be X minus two to the cube and then you can check this by taking due to all of this term, and it will become X minus two two to the power too. Um, so this is an figurative off of this. On the value at the boundary boundaries, X equals four minus X equals to what? Okay, So, Maxie Coast for this is the first one is sixteen over to which is eight on minus one third times. Mexico's a four four minus two is to to the Cube is eight. All right, so this is for actually close of four. Minus, actually close one, Max People's one. First term's just off one half. And second term, you know, it will be minus one third time's naked one to the Cube, which is ninety walk. So it's just a plus. My third. Okay, um, so we can calculate this. This here is five over six. Um, and this is three over eight. And this is twenty four or three minus. A overthe rate reaches sixteen over three. Okay, sixteen over three. And this minus ISS. Well, give us if a time, too. For the top and bottom. This is thirty two over six. So thirty two minus five is twenty seven. This is such a twenty seven over six. All right, this will be our answer.

And this question we're going to be going over how to find the area between those two curves. So what we'll do is we start with finding the intersection points. I already did a little bit of this here. But what you do is you equate the two curves to each other and then try and solve for X. I did so by squaring both sides and then just eventually getting a quadratic equation which I very easily solved for X. As shown here. Now, in order to figure out which is on top and which is on the bottom. What we need is we need to plug in a point within our integration interval and determine which has a greater value. And when I do this, I found that if I put in X equals zero, I had the square of three which is greater than the square of two, Which is greater than 3/2. That's pretty. That's going to tell us that this where is going to be the top function and the linear function? It's actually a line is on the bottom. Now remember that the area under between two curves is given by the integrating over an interval of the top minus the bottom, the whole thing integrated with respect to X. And since I have what is on top and on the bottom and our integration bounds, I can plug in all of this information right now. Remember that we found from 1-3 earlier? Yeah. Now I can split this into two integral because of the properties of the definite integral and I can From the second integral, I can factor out a 1/2. Now, in order to integrate the first interim, the first one, I will have to implement a substitution or I can you would let U equal to the square, you would let U equal to X plus three. And then you would just have to integrate the square root of you. And you would get new bounds. I'm going to demonstrate that here. I'll show it on the side. Yeah. So how we do this we will let you eagle X plus three. And do you will be D X. If I differentiate both sides with respect to X. Now When U equals negative when x equals negative three by plugging in -3 into X. in our substitution We'll get you equals zero. And similar logic When x equals one You will equal four. This means we're going to be integrating An integral from 0 to 4 of the square of you. Ew Now since you is to the one half power, I can just add one to the power and divide by the new power. So we will get you to the 3/2 Divided by 3/2 which is just multiplying by 2/3 Which will be integrated from 0-4. If I put in zero it vanishes and I'll just have to have you to the I'll have four to the three house power which is the square of four which is to To the third power which is eight And all in all. This will just give me eight times to over three which is 16 thirds. So now we know that this is that the first integral is 16/3. So with this over here this will be 16/3 minus. We'll have 1/2 times one half x squared plus three X. And that will be integrated from -3-1. And this will be multiplied by 1/2 by plugging in one will get 1/2 times one squared plus three times one. And we're going to subtract it with let's wrap let's nest this in a pair of parentheses to so this will be subtracted with 1/2 times negative three squared Plus three times -3. And all of that will be multiplied by one half. And all of this is subtracted with 16/3. And by evaluating this by doing all of the substitution is you will get that this is equal to 4/3. Uh huh. And that is the answer to this question.


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