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Sketch the region enclosed by the given curves and find its area.

$ y = x^2 $ , $ y = 4x - x^2 $

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Calculus 2 / BC

Chapter 6

Applications of Integration

Section 1

Areas Between Curves

Baylor University

University of Nottingham

Idaho State University

Boston College

Lectures

03:19

sketch the region enclosed…

01:01

Sketch the region enclosed…

05:00

05:08

03:58

03:20

03:08

02:23

04:10

Okay, let's get the region the first functions. Stand there. Two problems now. It just looked like this. Oh, pay up. So this is our Y cosa x square or a second function? Also a proble and we can't find find. Ah. Then why you go to zero X Either ecosystem for or zero across the region and for zero. Okay, this is our xx, sis. This is our by axis, since since the coefficient of X square is an active, you know, open don't downward. So there's probably something like that. Okay, So this is our why you coast to for X minus X square and the enclose regions red there by those two problems. So we need to find out the Intersect, those two intersection points and they already know they both cross the region, So we got one. So what about the other one to get the other one? He said as, um this point satisfy both functions, so you know, x square ecos. Why, Nico? So for X minus X squared. So we know X square on this point. Thanks. Thanks to all right. Well, do you know this point as x two so x two square should be close to y two and buy two ico. So for eggs, too. Linus x two square. And we can't get our AKs to from here. Our ex Who will be this? X two is not zero. We can cancel with blacks, too. For extra rico's too. Okay, um now, let's try to be better this area, since everything's represented the ex, we decide to buy the integral with respond with respect to ACS. So this integral and bad. And we got this DX and for bond. Dre, we just figure out it's just it goes from zero to two, okay? And the thing inside the injured girl is the upper curve, minus the lower occur. So hyper term being always for X minus. Expert, this is our op. Occur minus the lower curve. Which exact square. Mmm. So this then we need to you venerate this definite into girl. I'm too. I'd better with this. Definitely in the world. Um, it's just to find auntie duty of face. Um, and the beauty of this will be the first. Hermes just tow acts where? And this isthe minus two X, where if they combine those two together, so the ante Dios you with that ability to third execute and you can take devoted to check. So if I take my free take diverting with this term, this will become negative. Two three canceled checks. Where exactly? Match with here. So this is our auntie. Do it too. And then the benefit at the bone chri actually close to and X equals zero. We know Plaxico zero this thing Nico to zero. So we only need two. You'd better it at two. Next he goes to the first term is two times for, say, the second term here is to third times eight because to Cuba say this will be on eight over three.

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