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.
$ y = (x - 2)^2 $ , $ y = x $
Applications of Integration
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.