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

$ y = \sqrt{x} $ , $ y = 0 $ , $ x = 4 $

$(\overline{x}, \overline{y})=\left(\frac{12}{5}, \frac{3}{4}\right)$

Applications of Integration

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in this problem. We're trying to find Central plan. Soft region bounded by wise. You could skirt if x y zero and Mexico's four. So that is the shaded region. Officially, um, the center of mass should be somewhere around here. That is our estimate, Nellis. When the exact quarter, it's an ethically be no, not for you. First need to calculate the area that region an area we'll be in triangle some areas off that 10 strips with with dee eggs. So we're going to some of those up, meaning that wagon is integral. We're going to integrate this function f effects D x from X equals zero to execute us forces your four FX DX where if effects is credited, effects will be how integral a 0 to 4 skirted effects the ex Remember, skirted effects is extra t um one health. So it means that this integral will then be become to over three extra three over to where it changes between zero and four. And from this we see that then the area is it. Go to 16 or three. Now let's go clad. Ex bar export is one over area integral from A to B. where. A zero bees for X ffx DX No, let's play everything in the known at Area 16 or official one or area would be three old 16 inta go from 0 to 4 x skirted off x d x Dennis three over 16 Inter go from 0 to 4, thanks to the three over to D eggs the entire Garrett off. That would be X to over five extra to five. Forward to where exchanges between zero and four so we would find expire as three over 16 times to over five months. Fly by 32 which is four to the five. Forward to that is to sow The X coordinate would be at 12 or five in order to find a Y prime. We have are in Turkle as one over area again, which is three over 16. We know it from export parte integral from 0 to 4. Be designed How, What? Health Off the function squared where if effects as started effects. So skirted affects scribe D X that has three over 16 times one. Health is just constant, so we can take this one outside the integral integral from 0 to 4 x be eggs that is equal to 34 32 x skirt over to where I'd change it. T 04 From this, we see that we had three over 32 times 16 over too two, so there's three or four. So the central mess off the system is act on 12 45 and three or four.