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 - 4x $ , $ y = 2x $
Applications of Integration
Okay, So for this question, I also need to joy draw those curves. So here is my excellent access. And ah Mei Wai yee Coast to X square minus wax. It's a problem. And the intersection with XX is there will be zero and four. So here's across Cyril. And for, um, our problem look like thiss Okay, four and ah, it's the white goes to access is ah the use the line. So it's not that accurate bus. The idea is saying something like this. Okay. And the enclosed area will be here and because solve for this intersection point, you know this is Cyril. They're all across the region. But this you freeze. And this point, if we call this point x x two we can solve for X two. Um, let's sell X to first. Ueno acts too square minus four ax too. Should eco's too white too. It's just the height, the length off this this Sigmund and I will also no ax to satisfy this the second function So we know why To Rohit Coast Tio two acts too. And from here we can solve for X two. We know x two is not zero So if we divided backs to ample side, we'LL get X two Linus floor you close to two which means our acts to ICO to sex. Okay, so here six and since everything is they're represented by axe will integrate this Ah, with respect to X on da the tip co approximating wreck tango. So if we draw a typical wreck tech approximately right tango here Can I have a zoom out? It is the interview will be something like that. It's a rectangle and well, they still two x had the height will be a precursor of minus the lower curve after curvy is the line. So he's two x i minus the first car x i square minus four x y, which is plus or excited. So this rico's to six x I minus X I square. Okay, then we can ride out over formula for the integration, you know, area A rico's too Inge grow with respect acts. Ah, it will be the X at the end and the boundary for the boundary for acts will be zero and six. So it goes from zero to six and the integral will be six x minus X square six x minus X square. Okay, the next. Travis, evaluate this into girl. Um, you know this Rico's Tio? We find the anti devoted forthis, and he'd do it for this will be three x square, Linus Clan cube of one third x Q. Okay, and then the value. Add those boundary because six minus X equal to zero texting close to zero. And we know the X equal to zero. This is Cyril. So we just need to evaluate as six. The first term is three times thirty six, which is seventy two. A second term is up My third time. The six Q Ah, redivided. Sixty five three's too. So he's two times six square, two times six square is to talk. Okay. Oh, sorry. I made a mistake here. You see, this is not seventy two because we know X square. That X equal to six. X square is thirty six, thirty six times three. It's actually one know it. And the second term is one third times six is too, and ah, cancel the first sex with the three. That this will be two times thirty six, which is seventy two. Okay, otherwise they cancelled two zero really doesn't make sense. Because, as you can see, the area is now zero. So this no e codes to thirty six area of the shaded regions will be thirty six. Yeah.