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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 = \sin x $ , $ y = x $ , $ x = \frac{\pi}{2} $ , $ x = \pi $

$\frac{3 \pi^{2}}{8}-1$

Applications of Integration

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Michael M.

May 13, 2021

these are not the right questions that match the book

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Okay, so we need to draw those given curves first. Ah, a straw. Exploit access. Um, Then draw the first curve, which is Sai acts. Okay, so let's go see my ex Restart from negative pie. They so close to zero. Go on, stop. Bad pie only draw Lycos X is something like this. It's like this. Then tensions curve off saying X and coal region and ex eco's too compiler to here. Ok, Mexico's a pie or two. This is for the sign X You close to what? So here. No people. Um and, uh, this is extra coast by. Okay, so or the boundary of the region, there will be this shaded region right here s o r. Job is, uh you venerate this area. So how do we do it? We find approximating since everything we can see here is represented by ex So when they do the integral me too. It with respect to X. Okay, um really, really close to the integral. Always respect to x. So far so good. Um and I mean, you can fly in the boundary boundary for axes just goes from on spy or two party. Ah, And this here is just the upper curve on minus the Laura curve. Before that, we can see if we can Oh! Ah, Typical approximating rectangle. Something like that. It's very small here. If they do me come out, it will look like a test. Okay, this debacle approximating rectangle the bells will be tell tow x on how to be explain. Why, Linus. Sorry. Okay, so in other words, all right. The world will be pecs minus sorry, Lex. Okay. Yeah. He violated this integral to find area off the region. And did you ever give a piss? Is lor have X square and science ax and he derived table This's on an active co sign. Don't have Snoopy plus Cose i X It's a classical rex on the value Pickles pie, my nose Actually go to a pie or two. Okay. No, the first term. There will be twice. Where? Where to your second her co sign pie. You know, cosign. You know, coastline pies next year. What? So this is minus one? Um, can in my ass I was an accident. Close to a pie or two. This will be pi square or four times one have. So his price where or eight rum and coke. Oh, sign Hi over to is zero. So is plus zero. In other words, our area there will be high score over to minors that high score over eight, which is okay. Three pi square, over eight, minus one. Okay.