Question

Find the centroid for an area defined by the equations y = x² + 3 and y = - (x – 2)² + 7

Answer 1

The points where the 2 graphs intersect is  where x = 0 and x = 2.

-                 2                                  2
x   =    INT               x dA     /    INT        dA
                  0                                  0

INT dA   = INT -x^2 + 4x + 3 - (x^2 + 3 ) dx  =      INT -2x^2  + 4x
               =   -2 x^3/3 + 2x^2
               =   2.667  between 0 and 2

xdA  =   -2x^3 + 4x^2  INT xdA  =   -x^4/2 + 4x^3/3 = 2.667

centroid = 2.667 / 2.667  = 1  (x = 1)

Answer 2

The centroid for a bounded region is defined as:

$x_c = \frac{ \int\limits^a_b {x(f(x) - g(x))} dx}{ \int\limits^a_b {(f(x) -g(x))} \, dx} \\ \\ y_c = \frac{ \int\limits^a_b {\frac{1}{2}(f(x) +g(x))(f(x) - g(x))} dx}{ \int\limits^a_b {(f(x) -g(x))} \, dx}$
where
$f(x) = -(x-2)^2+7 = -x^2+4x+3 \\ \\ g(x) = x^2 +3$
limits are where f(x) = g(x) at x = 0,2
Subbing into the integrals
$x_c = \frac{\int_0^2 (-2x^3+4x^2) dx}{\int_0^2 (-2x^2+4x) dx} = 1 \\ \\ y_c = \frac{\int_0^2 (-4x^3+2x^2+12x) dx}{\int_0^2 (-2x^2+4x) dx} = 5$

The centroid is the point (1,5)