Centroid

Quote:

Originally Posted by viet

Find the centroid of the region bounded by the following curves $y = x^2, y = x+3$

Centroid : (___, ___)

I will assume it is a homogenous density.

In that case fist find the mass.
In this case simply the area.

$\int_D \int dA$
To express the region we need the intersection points between the line and parabola.
$x+3=x^2$
$x^2-x-3=0$
$x=.5\pm .5\sqrt{13}$
The upper curve is,
$x+3$ and lower is $x^2$ .
Thus,
$m=\int_{.5-.5\sqrt{13}}^{.5+.5\sqrt{13}} \int_{x^2}^{x+3} dy\, dx$
Thus,
$\int_{.5-.5\sqrt{13}}^{.5+.5\sqrt{13}}x+3-x^2dx\approx 5.67$ .

Now we find the first moments.

$M_x=\int_{.5-.5\sqrt{13}}^{.5+.5\sqrt{13}} \int_{x^2}^{x+3}y dy\, dx$
Thus,
$\int_{.5-.5\sqrt{13}}^{.5+.5\sqrt{13}}\frac{1}{2}(x+3)^2-\frac{1}{2}(x^2)^2 dx\approx 11.84$

$M_y=\int_{.5-.5\sqrt{13}}^{.5+.5\sqrt{13}} \int_{x^2}^{x+3} xdy\, dx$
Thus,
$\int_{-.5-.5\sqrt{13}}^{.5+.5\sqrt{13}}x(x+3)-x(x^2)dx=8.14$

Thus,
$\bar{x}= \frac{M_x}{M}=\frac{11.84}{5.67}\approx 2.08$

$\bar{y}=\frac{M_y}{M}=\frac{8.14}{5.67}\approx 1.43$