Answer
$(\overline {x}, \overline {y})=(\dfrac{3}{2},\dfrac{1}{2} )$
Work Step by Step
We have:
$Mass~of~area,~(m)=\int^{1}_{2} x^2 \times (2x^{-2}) dx=2$
Now, $m_x=\int_{1}^{2} x^2 (\dfrac{1}{x}) (\dfrac{2}{x^2}) dx=-2[\dfrac{1}{x}]_1^2=1 $
and $\overline {y}=\dfrac{m_x}{m}=\dfrac{1}{2} $
Further, $m_y=\int_{1}^{2} x^2 (x) (\dfrac{2}{x^2}) dx=[x^2]_1^2=3 $
and $\overline {x}=\dfrac{m_y}{m}=\dfrac{3}{2} $
So, $(\overline {x}, \overline {y})=(\dfrac{3}{2},\dfrac{1}{2} )$
