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Numerade Educator

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Problem 25 Hard Difficulty

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

$ y = 2x $ , $ y = 0 $ , $ x = 1 $

Answer

$(\overline{x}, \overline{y})=\left(\frac{2}{3}, \frac{2}{3}\right)$

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Video Transcript

in this problem, whereas to find be central mess off the shape formed by the line wise, you could do X lines y zero and X equals. One said that it would be to shape this shaded region and usually it looks like central Mass should be somewhere around here. Now, mathematically and articulate, less calculated. We first need to find the area off that shape and less first, then find the limits off integral. Since we're gonna take 10 strips with with DX and now we're going to some goes up, Um, we go from X equals zero X equals well, sort of limits. Infringe uncle would be 0 to 1. We have the function to x times DX that would be ex crab, where X goes from 0 to 1 in the area with them Meet one. You know that export is you go to one over area internal from a to B, where we know it to be 01 x times ffx d x What is several? Thanks. Well, this is our f effects wise equal to x. So we have one over one in general from 0 to 1 x times two x The eggs that is equal to two times X cube or three words changes between zero and one, so we can see that export is an equal to two over three. Well, for why bar we have are integral as one or area from A to B, where a zero bees won't we have one? How off the function squared the eggs that is areas one. So we have interviewed from 0 to 11 health. It affects his two acts. F of X court is four X squared D eggs that is then equal to these will cancel out He had two X cube over three or exchanges between your on one. And from this we see that why bar is all security. So the center of mess off the system list too earthy and two over three