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Numerade Educator

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Problem 31 Hard Difficulty

Find the centroid of the region bounded by the given curves.

$ y = \sin x $ , $ y = \cos x $ , $ x = 0 $ , $ x = \frac{\pi}{4} $

Answer

$(\overline{x}, \overline{y})=\left(\frac{\pi \sqrt{2}-4}{4(\sqrt{2}-1)}, \frac{1}{4(\sqrt{2}-1)}\right)$

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Video Transcript

in this trouble words to find a sentry doctor region bounded by Why sickle Sign X's and Y's ical co sign is X equals zero and executes Final Four. So that is the shaded region less for Scott played the area off that region. The idea is to form 10 strips with what d eggs and some does up between the limits off X. So since we're something we need in Tegel and exchanges, which means you're on fire before and we had two functions here, and as we can see our upper function isco, Sonics and Sign X is the lower function. So that will be FX and this will be dfx, so the area will then be FX. One issue affects T X or co sign. Next my sign X the eggs that is then equal to sine X plus Come sign effects where exchanges with means you're on fire before and from this we see that area is then skirted of two minus one. All right, let's Cochran export. We not an export is one over area. In turn, go from a to B X times if effects minus Jeff eggs the eggs and we know that Why? Bars up for one over area and Turkle from a to B one. Health off F scarred effects might sti skirt if x the eggs. Okay, we have everything that we need. So let's display. Those in expert will be no Niko to walk over. Skirted of two minus one times. Ingi go from 0 to 5 or four. Thanks. Times come. Side effects minus Well, side effects the eggs that is one over. Skirted of two minus one in jungle from zero X pi over four. Ex co signer. Thanks. Minus, um ex side effects. The eggs. We're gonna use integration. What parts? For these two terms. So for the 1st 1 we're gonna say that X is equal to you. So d X would be good to do you and consign effects. D X is equal to TV, so we would be good to sign of X. That is, for the first part. For the 2nd 1 we're going to say that X is equal to you. So sign X uh, d X would be TV if X Is he good to you? D X is sick with the DEA and negative off coast zannex physical to be now performing integration. What parts we got Explore as won over Skirted of two minus one X sign X Plus Co. Sign X minus sine X minus X Co Sign X, Where exchanges between zero file before plugging in pile before and zero and doing the calculation we find export to be equal to pi. Times. Skirted of two minds for the island by four times. Started off too much. It's one we also need to track. Wait, why bar? We're gonna use the formula that we right before so we own at why bar is equal to one area internal from zero fire or four one. Health off if skirt of X minus G's for eggs D X. Let's just like everything in that is you could do one over skirted of two minus one that is one over area. One of us is just a constant so we can take this one outside. Integral. We have one health off, ain't I? Go from 0 to 2 pi over four cosas Cardiff X minus Sign squared off X d. X Welcome Subscribed effects when science card effects is equal to co sign to access coastline to X and we condemn, right? Why prime as one or two times skirted off to my sworn uh, signed two ex delighted by two were extensive. Means you're in power for sent to have it two extra wide apart, too. Is an entire derivative of course, side to X and plugging imply before zero we have one over four times. Skirted of two minus one. We got four by lots playing two and two. We have one minus zero. So from this we see that. Then why bar as at one over four times skirted of two mice, one for the center off mess off the system will then be that pi ce credit of two minus four, divided by four times skirt of two minus one and one over four times carted off to minus one.