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Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y^{2}=x, y=2,$ and $x=0$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 4

Centroids

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we want to find the area of the region bounded by why equals two minus the absolute value of X and y equals X squared. So we can find the area of this piece right here and just double our answer, since the graph is the same on both sides and that will give us the this total shaded area. So it's first figure out at what point these curves intersect. So I set X squared. You quit too, minus X. And if we move everything to one side, we get X squared plus X minus two. It was zero. And if we factor, we get X plus two times X minus one equals zero, and so X equals native to and X equals one. And since we're talking about this point here in the first quadrant, it's the X equals one. So are integral will go from 0 to 1, and it will be to minus X minus X squared. These absolute value signs, uh, can just drop off because we're dealing with the positive X is in the first quadrant. So are integral is going to be two x minus 1/2 x squared, minus 1/3. Execute from zero. The one. And if we plug in one, we get to minus 1/2 menace 1/3. And if we put in, Syria will just get zero. So we're left with for over two. Mina. Sorry. Uh, six over 12. Over. Six, minus 3/6, minus 2/6, which is 12 minutes. Three is 99 minutes to a 77 over six. That is our area of have of the shaded regions. Our total area bounded by the equations will be 14/6 or 7/3.

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