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Sketch the region enclosed by the curves and find its area.$$x^{2}=y, x=y-2$$

$$\frac{9}{2}$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

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In mathematics, integratio…

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Sketch the region enclosed…

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04:47

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03:23

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02:13

uh, not sure where they wrote the problem this way. Um, but y equals X squared. Looks like this to probable. I think everybody knows that if you're in, if you're in countless, um, and then the other function X equals Y minus two. Um, I write as why equals X and you have to add the two over, Um, so, like, shift up to and the soap is up 1/1 line like this. Now, here's the issue is we're definitely find in the area between these two, but we don't know those bounds like I don't know this x value in this X value question mark Question Mark. So you find that by setting them equal to each other, X square is equal to X plus two expo area and subtract X. I'm gonna subtract two over. I can look at this as a quadratic that a factors A to, um one into that had to be native one. Well, that works of it's plus one minus two. So I get X plus one next minus two, which means my bounds and this should make sense is negative one and positive two. So if you plugged these and you'll get the same. Why values A one plus two is one. They won't square this one. Two plus two is 42 squared is for okay, so don't checks out. So those are my bounds from native wanted to. My upper function is the X plus to function and you need to subtract off the lower function, which is the X squared. And so you find area The X man From here, it's just the anti derivative. So the anti derivative of X, you add one to your exponents and then you divide by that new exponents plus two x again, you had one to your exponents and divide by your new exponents. You can double check the derivative of this needs equal this so and it does You just believe me from negative 12 tubes. And now we need to plug in your bounds. So, you know, plug in to when you square it becomes four. Two times two is four. I'm plugging into, by the way to Cuba's eight thirds and a minus. When you plug in negative one into all these negative one. Once you squares positive two times able in this minus two and then a one cubit is still negative. One so changed that to be close one third. Now, from here, I would go to a calculator cause you're gonna get some fractions and just you can go with me as I'm this left wipe in parentheses here is 13th. Um, and when I get on this right from parentheses is negative. 76 And so going back to the 13th minus negative 76 you get an answer of nine house or 4.5. I like fractions.

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