A plane figure is enclosed by the parabola $y^2 =4x$ and the line $y=2x$. Determine the position of the centroid of the figure.
Here is what I tried :
Plotted the graph.
$y=4ax, y=2x \\
4x(x-1)=0 \quad x=1, x=0\\$
$\bar{x}=\frac{\int xy \mathrm{dx} }{\int y\mathrm{ dx} }$=
$\frac {\int ^1_0 x^{\frac{3} {2 }} } {\int^1_0 x^{\frac{1 } {2 }} }\mathrm{dx} $=
$\frac{ \frac{2} {5} x^{\frac{5 } {2 } }} {\frac{2} {3}x^{\frac{3 }{2 } }}$ $\Biggr|^1_0$=$0.6$
etc
Where am I getting it wrong?
Best Answer
$$\overline x=\frac{\displaystyle\int_0^1 x(\sqrt{4x}-2x)\,dx}{\displaystyle\int_0^1 (\sqrt{4x}-2x)\,dx}.$$