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

$ x = y^4 $ , $ y = \sqrt{2 - 1} $ , $ y = 0 $

area $=\frac{22}{15}$

Applications of Integration

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uh, okay. Sketch the region closed by the given curves and finance area. So first, let's draw those two curves. This is our XY plane. And, uh, the first function is X equals. So why to the power floor? So it looks like a problem, but it grows much faster. So it opens to the right like this. This will be my ex seacoast y to the power four. Oh, and then the second term here, Um, why he goes to the square root of two minus x. Yeah, this we know. Oh, yeah. Um, it can be represented as if we take the square. This is just a wide square equals two to minus X. We move around things. This is equivalent with X equals two two. Wine is Y square. What? Let's score. Yeah, and, uh, be careful here. We also need this indication why is greater or equal to zero? Because you know, the square root my equal to the square root of something, So why cannot be negative? Yeah. Okay. And this is also a problem. He opened to the left on a start at two. And open to the left. Okay. They will be enclosed area will be here. Okay. And here is to Here is General Oh, uh, and as we can see, um, since we reformat the second curve right now, everything is represented by why. So if we want to find an area and take the integral with respect to why, instead of facts and and the boundaries at this point in this point, So if I do know this s by one Yes. S y two how to find them? It's just, um Oh, be careful here. The enclosed region is actually now the entire thing, because we could this extra condition here when we reformulate the second curve, why should be no negative. So, um, this enclosed area is only this half of them is here. Here is empty. Okay. Um, therefore, we don't have this by one. We only have this fight to on our right. One will be zero. It goes from zero to white too. Okay. Mm hmm. So now we know. Mm. Our boundary by one will be terrible. And why to surely satisfy in a way to is positive. So what? You should satisfy two minus. Why square the second curve and the first curve echoes too wide. The power of four. Okay. Mm. Uh, as we can see, if I plug in by coast to one. This equation hopes so you can say are way too. Must be one. Okay. Or you can do the factoring Then you can solve for this, uh, um, polynomial equation, you'll find your roots, and the positive route will be our right to, and it has to be one. Um, So the boundary will go from 0 to 1. Um, seeing inside the integral is the record minus left, Curve like curve. Here is just two minus. Why? Square minus the left. Her left her is, uh, right to the power of four. An anti duty of that is to why mine is one third like cube, minus month. Faith, my faith and inveterate one and zero and receiving Michael 20 this whole thing you go to zero. So we only care when y you go to one plug in white. Go to one. We got two minus one third, minus one faith. So it will be on two minus one third. My nurse on fifth. So this this is, uh, 8/15, right? So it will be 30/15 miners, eight or 15. Oh, So this whole thing will be, um, 24. 22 over 15. Mm, 15. Okay.