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Numerade Educator

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Problem 36 Hard Difficulty

Use Simpson's Rule to estimate the centroid of the region shown.

Answer

$(\overline{x}, \overline{y}) \approx(4.4,1.4)$

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Video Transcript

in this problem, we're given a graph and worse to approximate The Central Office given Region using seems since rule You know that eccentric oi is equal to summation of Ex Terms area divided by total area and that is integral from a to B x times d, uh, function the X divided by function the X and why part is equal to integral from A to B so the limits of the region Times one Health function square D x Divided by function in times d x Here we know that the limits are from 0 to 8. Um, instead of using this exact integration were Guineas, Simpsons Rule and these are deployments that we're gonna all that dysfunction at most after points are just approximations. You could use different values. So let's start by approximating the functions in the denominators are inter goal FX dxL Let's start our that interval of effects The eggs a CZ You can see the increment between each point is one. So we're gonna set up for Simpson's rolled out the extra p want and we have in total eight numbers. So end will be eight cell for The Simpsons roll If X T X will be 1/3 fo zero plus four ash off one plus two effort to four F off three plus two F before plus 4 +45 plus to f off six. It was four f of seven plus two f ate. Sorry, actually, for Fayed is the last one. So we're not gonna have this constant here. Settle Just people. It's a fade. We know the functions values at those points. So we're just gonna plug those numbers in, do the modifications and doing that, we found this integral approximation as 20.1. Now, let's step works, mate. Um, the numerator for ex enjoy. So x times, FX, the eggs Don't we go to 1/3 all we're gonna have the same thing. All the differences were discontent and multiply it by the X values. So it will be zero times fo zero plus four times long times that one plus two times two times that off too. Plus four times three times F three plus two times four times before plus four times five times five plus two times six times out of six. 47 F off seven and ate a fade. If he again plugged values in from this craft. And if you do the multiplication, we see that X ffx. T x is about 88.13 Bilis Brooks made integral What? Have off f squared x d x We're gonna have What? Health isn't constant. 1 30 comes from the same central. We're just gonna know, um, square to function. So we're gonna have and scared of zero plus four times that script of one plus two times x squared of two. Plus that that out. You get the idea. We'll have four times subscribed of seven, plus, um, f squared off eight. Now again plugging into function values. He found this approximation to be about 29 points here. Six from missile being introduced, no to export. And why bar? We know that export is equal to this approximation divided by this approximation. So that is, uh, 88 0.13 divided by 20.1, and that comes out to be 4.38. Therefore, wire bar, we have this approximation divided by Brooks mission for the area. So 29.6 divided by 20.4. And that comes out to be 1.44. So the center of mass off this region as for 0.38 then 1.44.