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

$ y = 12 - x^2 $ , $ y = x^2 - 6 $

72

Applications of Integration

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All right. So let's catch the region. First. Say, this is our exit. My plane. And our first function is twelve minus x. Where, uh, so it's a proble, um, open downward. So but acts of, um X equals zero r Y e cosa twelve. So twelve here. And but why go to zero or x square? Kosher twelve. Which means X equals two, two times square, Route three minus, plus and minus. And based on based on this three points, we can draw her problem. Look like kids. All right, on this. Why? Because tow X square minus six. My x equals zero Baikal Connective sex. Yeah, my code zero actually. Produce two square root of six. Okay, the hair is in that cave skirt off. Six. The problem. They looked like thiss. Okay, so here the shooter region will be our target. Okay, Um, as we see, everything is represented by axe. So if we wantto integrate this region, I used to go to find the area off this region. We just need to find We need to take the integral in terms with respect to X. So we want to take the integral this wrist back to X. If I write down the formula a very close with the inch girl he X then that's fine. The boundary for X, The boundary as we see it. Cemetery. Um bye. Why Exes? So I say, Here is next essay. Here is X one. Here is X two An xbox too. Just intersectional those two curves. In other words, Xbox, you should satisfy twelve minus x square. Rico's tow X square minus six. So if we move around things this will be two acts were very close to eighteen. So x square, very cosign. My accident goes to party here and ninety three. So our X one recall to ninety three and Axl goes to three. The boundary for X goes from negative three to three. Right. And the things either into girl, it will be the opera car minus the lower carb. So our pet crow here is twelve minus X. Where and the lower curve here is just X square minus six. Okay, so this will be to grow from negative. Previously on a king because twelve plus six is eighteen minus two X square gx Okay, on an anti duty for the inside is Excuse me uh eighteen x minus two third, Execute any battery at me on ninety three. This whole thing. So Oh, Eco's too are executed. Sway, this is forty five. Another fifty four. This is fifty four, huh? Mine is so three Cubans twenty seven or three. This ni so minus eighteen in the minors. Plug me in ninety three B negative. Fifty four and ninety three. Q base nineteen twenty seven. That connected. Cancel this. Plus two. Third twenty seven is wass eighteen. Right? And this eyes, um thirty six here is naked thirty six minus thank you. Thirty six. So the whole thing will be sent to