The polynomial $f(x)$ satisfies the condition $f(x + 1) = x^2 + 4x$. The area enclosed by $y = f(x - 1)$ and the curve $x^2 + y = 0$ is

Find the area lying between the curves $y^{2}=2x$ and $y=x$.

The area included between the parabolas $y^{2} = 5x$ and $x^{2} = 5y$ is

The area (in sq. units) of the region bounded by the curves $y=x^2$ and $y=8-x^2$ is

If the area of the region $\{(x, y): |4-x^2| \leq y \leq x^2, y \leq 4, x \geq 0\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right)$,where $\alpha, \beta \in \mathbb{N}$,then $\alpha+\beta$ is equal to . . . . . . .

The area (in sq. units) of the part of the circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)$ where $\alpha, \beta$ are coprime numbers. Then $\alpha+\beta$ is equal to . . . . . . .