Using Green's theorem find line integral $\oint_C (-x^2+x) dy $ enclosed by $x=2y^2$ and $y=2x$

Question

Using Green's theorem find line integral $\oint_C (-x^2+x)\, dy $ enclosed by $x=2y^2$ and $y=2x$

The intersection points between the line and the parabola are

$$P_1 = \left( 0,0 \right) \quad P_2 = \left( \frac{1}{8} , \frac{1}{4} \right)$$

That said, knowing that

$$\frac{\partial Q}{\partial x} = -2x+1 \quad \frac{\partial P}{\partial y} = 0 \qquad \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = -2x+1$$

So

$$\oint_C (-x^2+x)\, dy = \int_{0}^{\frac{1}{4}} \int_{2y^2}^{\frac{y}{2}} \left( -2x + 1 \right) \, dxdy = \cdots = \frac{3}{640}$$

The answer of the textbook is different.

Textbook's answer: $\frac{1}{5}$

Did I make a mistake somewhere?

Answer 1

You changed $-2x-1$ to $-2x+1$ in your last integral.