Centroid of plane areas

**jasonlewiz** · March 3rd 2011, 06:40 PM

Can someone show me the solution of this problem? Thanks..

a.Find the centroid of the area bounded by the curve y=x^2 and the line y=2x+3.

b.Find the centroid of the area bounded by the curve y=2x+1, x+y=7 and x=8.

**HallsofIvy** · March 4th 2011, 04:26 AM

No, I won't show you the solution, but I will show you how to solve it yourself. If you were given a problem like this, you are presumed to have learned the formula for coordinates of a centroid:
If $(\overline{x}, \overline{y})$ is the centroid of region R then
$\overline{x}= \frac{\int_R\int x dA}{\int_R\int dA}$
$\overlne{y}= \frac{\int_R\int y dA}{\int_R\int dA}$

where "dA" is the differential of area for the region (so the denominator is just its area), dxdy in the case of Cartesian coordinates as here.

In particular, for the region between the graphs of $y= x^2$ and [maht]y= 2x+ 3[/tex], the centroid is given by
$\overline{x}= \frac{\int_{x=-1}^3\int_{y= x^2}^{2x+3} x dy dx}{\int_{x=-1}^3\int_{y= x^2}^{2x+3} dy dx}$
$\overline{y}= \frac{\int_{x=-1}^3\int_{y= x^2}^{2x+3} y dy dx}{\int_{x=-1}^3\int_{y= x^2}^{2x+3} dy dx}$

In both cases, the denominator reduces to
$\int_{x=-1}^3 (2x+ 3- x^2)dx$

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