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

$ y = \tan x $ , $ y = 2 \sin x $ , $ \frac{-\pi}{3} \le x \le \frac{\pi}{3} $

$A=2-2 \ln 2$

Applications of Integration

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Okay, let's try to sketch the region close by those two curves. And since the domains by a specified only, I need to call the curve inside That dont make so our export. Here's our x y plain. No, this is good. Okay, look said this It's over. X y plain or ex goes from negative power three to power through it. Okay. And so of X equals power three. Tension to X equals two squared three No, the ex ecos inactive power three tens Next because through back to square root. Right, Except with zero A zero and lee no tension axis at our function. So it looks like this. And for on the other hand, for the second her are two science, you know, sex look like this so we don't We need to find the end points so and points is over X equals to power three. We know science is screwed three over to have a book Piper too. So it will be square through it assed we can see it was to curves Army intersect Had those toe end point who saw a region because Tucson accidents also our function so it will play it right here here is already close the region. And as we mentioned, everything here is symmetric by this region. So we know those two parts has I have exactly same area. So we only need to calculate one off them. The more diaper too. So say I want to calculate this area. They will be integral with respect to X. Since those two curves can be represented by X and the up the boundary here is goes from zero to three power suite and things have been integral. Will be upper curve minus the lurker, which is to sort of fix Linus tension Tex okay. And instituted off this So just mine is to co sacks minus the art. Andi, do it for ten years. We know that anti do it. If attendant attention access Cave lock actually cause annex. So this hoop clause log Who's Alex? They value it at power of three two zero so well, actually cause power three. This is over half. There's just two times previous form plus log minus X equals material is it's two times conectiv to here because cause and zero is one And we know Logue y zero so minus two plus zero. This will return Here is a plus for minus two is two and a plus two times log over too. Bye. And no for this log, we can bring one bag to sign to make this love too. What I'm saying? Here we have this for a general log. It's like negative bog A over B. Did he close to lock Good or anything? All right. It's like that. You re bringing in the negative side. Take the reciprocal inside. So this were also he close to two minus two. Block two. Yeah, We're both too can be our answer.