Problem 19 Medium Difficulty

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

$ y = \cos \pi x $ , $ y = 4x^2 - 1 $

Answer

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Video Transcript

So you need to sketch the region close by. Those foolery in curves r say this is our x y plain. And so the first function is cosign packs that in some ex ico So wan Whose empire? Ecos huan Negative one so and we're accident post tow her half cose AIPAC seacoast Terrible. The maxi code zero coasts and zero is pie. Couldn't Ciro is one all right? People of this function should look like and this is a union function. So is symmetric by y axis look like this. It's actually cause one. This is not your war. Excuse me. This is our Mexican snack you want and Lycos connected in the second on the second curve. It's a problem. No X equals zero. This curve across a bike was naked one the waxy Sarah Why post make it one across this point? And then why could zero no ax eco's to a positive and that give by half So I call it suppose to quite this make you happy This problem back ten okay. And then close regions in between this shaded area? No. So you orderto finance area. We write on the intercom role for area ico swindle grow with respect, Teo Truce acts because those functions can be represented by acts explicitly. So this is a coast too integral with respect of the base, Back to backs and X goes from negative or half to part with him. In here you are. It's the upper curve minus the Lord coz I pax miners blacks were right, That's what. So it will be minus wax square plus four All right, next to the value of this indefinite things grow thiss definitely grow. So it's fine. And he knew it. The first term will be over. Pi sign packs in the second term will be, um for over three. Execute a third Who just x Yeah, you value? Uh huh. No. So the first time that calm Craxi coast, where half the no sign pi over two is one nice world Were pi miners for three plus one Miners plugging because inactive half So as you can see those are odd function Therefore everything flip their signs So when you hear is the first time we'LL be inactive What? Have, uh thank you for our pie But sorry been mistake We're plugging X equals one half here So this is not forward three times what this is for over three times one over eight. Now here is just And plus we're half right in the second term. Just sleep the sign. So the second This is Manas. Where? Over pi and plus for three times. Well, eight. Linus, where Right. And since we have this negative sign for bringing in our thing, we'LL flip again. Therefore, we'LL get two times the first first herbs. We only need to calculate the first term. And what flight by two. Right, So here will be over six. Plus my half That's our three. So he's just too or pie last to third.