Problem 12 Medium Difficulty

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.

$ 4x + y^2 = 12 $ , $ x = y $

Answer

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Video Transcript

So again we need to find Ah ah. We need to draw the curves given curves. And this time, um, it's not very clear The function is real ize It can be represented by X or why so we reform this first formula here and we see we can see the we can't express acts in terms ofthe why very easily. We just moved. Why square to the other side? I'm divided by four. So it will be twelve miners. Weiss. We're over four and ah, this is just ex Pecos way after reformulating we can see that we can say everything is represented bit by Why? So why don't we take the integrate Made the respect Why? All right, um, let's draw the pictures. So here is X Y access. You know, if I draw this, this is a problem. Um, then why he called zero X equal to three. So I just look like this. And a x Pecos to zero. Why he caused Teo tough. So why will Why square very coast twelve. So? Well, he close to, uh, two times Square three and negative. Two times Square three. So this problem will look like this. Okay, and I also got this wiII coast packs It's the street line And I would go those two intersection So this is X coast boy. Um and I miss Ah, no time for this. Do you know this intersection X one here. This intersection is X to here and corresponding x one corresponding to buy one and ah, actual corresponding to bite you since we decided to integrate with respect. Why less fine hour? Why want wide too? All right. Here is the enclosed the region, right? No. You know to find the boy y y too. Ah, you know he should satisfy this equation. Twelve Oh, minus Why square over four Pecos. Why, that's when those two curves intersect. So this will give us with times for on both side and move everything to the right. Give us a boy square Plus for why minus twelve. He called to zero and we can solve this a factor. A factoring So this is a live plus six. Why? Minus two You go to zero and we know why wise the smaller one. So where wass close to ninety six and why to coast to Okay, um and we draw tape approximating or tango since we decided to integrate with respect. Why are right? Tango looks horizontal, right? Say, over typical rectangle goes here If we zoom out there look like this home and here is dealt Why, um the the the heart ofthe this rectangle will be the right carved minus the left curb. So the right curves right now is this whole thing. So you will be twelve, Linus. Why? Hi square over four Mari this y I Okay. They're, uh, next page in to find the area of the shaded regions. I did note as a so are able eco's to the integral with respect to why so have the y at the end and the boundary for why goes from it goes from my one two y two. We know why y's negative sakes and lie to you goes too. So it's goes from ninety six two two and integral in sidings her into grow is just ah, this height off approximating rectangle. Without a subscript it will be Charles minus y square over four minus. Why? No. In the end, he derivative of that will be first Hermes twelve or for his way. So it's three y and a minus Ah, Y square over four. Anti akira tip of that. It will be of why over tough time's white cube minus Why over too. Why square and this evaluated at the bank three hiss. Why comes, too? Yeah, by coast Negative six. This very coast tio y close to this is six miners two Cuba is eight. Eight or twelve is two or three miners. Um, to square, which is for over two is two. Hey, this miners, But plug in white coat tonight of six. This is nineteen. Eighteen, um, minus more over itself. Times negative six Q and minus. But I have times thirty six riches and him. All right, so this really coast Tio, um, let's look at this time first, this cancer is one six. So it's one half. Oh, the negative sign canceled. So this whole term will be positive. Um, Isla have times, so it will be positive. But I have times six square, which is thirty six. So plus on a thirty six divided by choice is placid. Okay, so those eighteen cans old God, this is for minus to third, minus negative. Eighteen. So is class eighty. So this whole thing will be twenty two minus to third. Twenty two minors to third of the sixty six over, three, minus two third. So our answer there will be sixty four over three.