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Numerade Educator



Problem 24 Hard Difficulty

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

$ y = \cos x $ , $ y = 1 - \cos x $ , $ 0 \le x \le \pi $


sin $=\frac{\pi}{5}$ $=\frac{\sqrt{3}}{2}$ sin $(\pi)=1$


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Video Transcript

Okay, So the first part of this problem except asked us to sketch the region enclosed by these given curves. So we'LL start with that. Okay, Here's where access will look like we'LL start with co sign of X and this clear familiar with So just mark down some key points we'LL start with I ever hear And then high over too Okay, so then you know Cem values of co sign for example A sign of zero zero equals one co sign should know either. Well, that sign of pirate too equals zero And finally, Mia Cosa Hai, and that equals negative one. So we're gonna use these values Toss of help us Dar graph for the other function as you'LL see. So first we have these values So we can What? Grass? Yeah, And this and we know the general shape of the coast line graph. So look something like that. Okay, Now we want to draw our graph of one minus co sign of X. And since we've already figured out these values, we can already fill in most of it. So for one liners, co sign of X, we have zero. Yes, at high over too that that would be one minus pirated one minus one. Because the next one that is this and finally at high. Shit. Hi. We have that. It's one minus negative one. So that is two. And I specify that. Negative there. This is too SNL. It's villains graph. And again, Think about what this graph should look like. Well, it's the coastline graph, but flipped. And then it shifted up one. So this general shape will still be that of a co sign graft. They look something like Okay, I just want to emphasize the curves a little bit more, So I will redraw this. It's lovely. Okay, so now that we have the graph sketch, we want to find the area of this region over here in this region. Okay, so this means we have to first figure out what this point right here is because you have to integrate from here to here. And then we need Teo switcher graphs and integrate from here here between the first calculate where these two graphs intersect. So let's do that. We have this gives us partner equals money was too close an axe or what has if coz next they remember that X goes from zero to pie. So the only solution we can use are possible values from Europe. I So, since that's the case Oops. So that's the case. You know, the coastline of exes. Pirate three. They're not good. We know access people privately. This makes costar Vex. He could want of it. So now we know generally where into role look like so from zero to pie. Over. Three have co signs on top and one minus. Co sign is bottom surgery. Subtract. You know, bigness attract, um, one minus co centerbacks from co signing checks. Then from high over three to pie, we're going to subtract coastline of X from one mind to co sign cheques. So on the next page. All right, the integral. So we have now. It is fixed in front. He said it looks cleaner. I'LL try squeezing the DX over here. No, not that, actually. All right. Right now it's just a matter of simplifying the integral ls that we have? You are. You could I'm sorry. She relies on us to the next my sight. Genesis high over three one, right. She cooks and right there and then integrating this gives us and into a girl. Of course. I kn ends. She sign. Okay, we're almost there. So remember, pirate So evaluating Sinan Pirate three. It's just Let's just write a some common values of sign. So weaken wisdom our calculations. So we have actually have never. It's right here. So side note signs, however, three equals mystery over too. And sign pi there was one. Okay, So now what do we have? We have minus three and then plus two signed pirate Three, three, three. Hey, now we value is zero and I forget to mention what signers ears. But they just couldn't do that. So we get minus zero for mind sex here. Maya's Linus, you're technically didn't matter. And then minus two sign of zero zero. So this part is irrelevant. So it was just giving them. Okay. Okay. Now back to the first Interpol we have for the ten second grocer. You have what? Hi! Minus two signed pie to sign. Pi is minus two plus pirate three. Maya's to sign Piratey. So that b minus reading. Okay, so putting this all together, we see that something's cancel out. They have this cancels out of this. I can't serve this so we're left with. Our final answer is pilots too