Answer
$(\overline {x}, \overline {y})=(\dfrac{33}{85},\dfrac{698}{595} )$
Work Step by Step
We have $\overline {x}=\dfrac{12}{17} \int_{0}^{1} x(2-x^3-x^2) dx \\ =\dfrac{12}{17} [x^2- \dfrac{x^5}{5}-\dfrac{x^4}{4}]_{0}^{1} \\= \dfrac{12}{17} \times \dfrac{11}{20}=\dfrac{33}{85}$
Now, $\overline {y}=\dfrac{12}{17} \int_{0}^{1} \dfrac{(2+x^3+x^2)}{2}(2-x^3-x^2) dx \\ =\dfrac{6}{17} [4x- \dfrac{x^7}{7}+\dfrac{x^6}{3}-\dfrac{x^5}{5}]_{0}^{1} \\=\dfrac{698}{595}$
So, $(\overline {x}, \overline {y})=(\dfrac{33}{85},\dfrac{698}{595} )$
