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

Area bounded by the curve $y = \min \{\sin^2x, \cos^2x \}$ and $x-$ axis between the ordinates $x = 0$ and $x = \frac{5\pi}{4}$ is

If $A$ is the area bounded by the curves $y = |\cos x|$ and $y = 5 - \frac{4}{\pi} |x - \pi|$ in $x \in [0, 2\pi]$,then the value of $\left( \frac{A}{2} + 2 \right)$ is

Let $f: R \to R$ be a function such that $f(x) + 3f(\frac{\pi}{2} - x) = \sin x, x \in R$. Let the maximum value of $f$ on $R$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta > 0$,is $\alpha^2$,then $30\beta^3$ is equal to ————

If the area of the region ${(x, y) : -2x + 1 \le y \le 4 - x^2, x \ge 0, y \ge 0}$ is $\frac{\alpha}{\beta}$,where $\alpha, \beta \in N$ and $\gcd(\alpha, \beta) = 1$,then the value of $(\alpha + \beta)$ is:

The area enclosed by the parabola ${y^2} = 4ax$ and the straight line $y = 2ax$ is