Answer to Question #315670 in Calculus for gene

Question

find the centroid and boundaries y=x^2 and the line y=x

Answer 1

$S=\int_0^1{\int_{x^2}^x{dydx}}=\int_0^1{\left( x-x^2 \right) dx}=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}\\c_x=\frac{1}{S}\int_0^1{\int_{x^2}^x{xdydx}}=6\int_0^1{x\left( x-x^2 \right) dx}=6\left( \frac{1}{3}-\frac{1}{4} \right) =\frac{1}{2}\\c_y=\frac{1}{S}\int_0^1{\int_{x^2}^x{ydydx}}=6\int_0^1{\left( \frac{x^2-x^4}{2} \right)}=3\left( \frac{1}{3}-\frac{1}{5} \right) =\frac{2}{5}\\\left( \frac{1}{2},\frac{2}{5} \right) -centroid$