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.
$ x = 1 - y^2 $ , $ x = y^2 - 1 $
Applications of Integration
right for this question. First, let's draw the given curves. Here is X Y access and ah, this ex seacoast one minus y square. So we know why. ICO to zero ex Pecos What so equals two across one zero this point here's more. This is the coefficient before Why is negative? You know this problem And it opened to the left. So look like this, um, forever why you go to the other X equal to zero. We know why I called it a ghost to one ninety one. Okay. And this it's, um, very symmetry, um, to the first function, you know, And like a zero dysfunction coast to ninety one. And since the coefficient ofthe white square is one is positive. So it opened to the right and, um, when x equal to zero it by also eco's one ninety one. So it's in the office, all right? And they cook in close region and sit in the middle here. Okay? And for this question, as weekends, everything is represented by why, right? Our ex can be represented by why thiss the first curve is represented by y in the second car. Oh, so represented by why so we decide to integrate with respect to why. And ah, Leslie want to integrate with the respect. Why the typical approximate Here red candle will go some horizon's Oh, Why's that? Because they want Howard d y with which is sort of delta y here. And why goes Mari Ko so dealt? Why is here? So the typical approximately right Greg rectangle will goes horizontal. Okay, so if we zoom out this tico approximating rectangle, it will look like thiss. So here is our delta. Why? And the height, the height. Say this point. It's ah Why I, uh, say at this point this what I Then the height will be the left curve minus the right car. The right card minus the left herb. So it will be one minus. Boy, I square. This is a right curve, this point minus why I square minus four. Okay, so this will Rico's Teo to minus to why I square. Okay, um there Nice try t o find the area of this region. You know, a well eco's to the into girl since they take the inte gra integral with respect to why, you know and the end, it'LL be D y It's like or doubt why here we know our why goes from ninety one to one. So it goes from making one to one. We're here. We're just copy down this height when she's to minus to war. Where Right, Um Then let's find Auntie directive off. This death thing on the first term is just too. Why. And a second term is to third. Why, Q? Okay. And the boundary will be Wide Coast Juan. And why I chose to negative. Or, um So this eco's tio right here It's a really ICO. So why cause one? This is two minus to third and ah, minus on. Why close to ninety one, which is negative. You too. And, uh, thank you. Want to the Cubans ninety one, next to signs. Cancel With this minor side. So is a plus two third. So basically, we got four minus for four miners for three. So it will be eight over three. No,