1. Centroid problem

Find the centroid of the area bounded by the curve
$
2y^2+2y-x-2=0
$

and the line $x=2y$.

2. Do you know how to double integrate?

$x=2y^2+2x-2$
$x=2y$

$\int \int ~dA$

Using $dA = dx~dy$ will be better. It'll be easier to apply the bounds for y, because the functions are y's functions.

A = $\int \int ~dx~dy$

M_x = $\int \int y ~dx~dy$

M_y = $\int \int x ~dx~dy$

$\bar {x} = \frac{M_y}{A}$

$\bar {y} = \frac{M_x}{A}$

And the centroid is $O(\bar {x},\bar {y})$.

Now apply the bounds and integrate and so on..

3. Here's another graph I made