For $0 \le x \le \pi ,$ the area bounded by $y = x$ and $y = x + \sin x$ is

Let $f: R \rightarrow R$ be a twice differentiable function such that $f(x + y) = f(x) f(y)$ for all $x, y \in R$. If $f^{\prime}(0) = 4a$ and $f$ satisfies $f^{\prime \prime}(x) - 3a f^{\prime}(x) - f(x) = 0$,$a > 0$,then the area of the region $R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}$ is:

Find the area of the region bounded by the parabola $y=x^{2}$ and $y=|x|$.

Let $f(x) = \min \{\sin^{-1} x, \cos^{-1} x\}$. Then the area bounded by $f(x)$ and the $x$-axis is:

The area of the region bounded by the curve $y=4x-x^{2}$ and the $x$-axis is

The area bounded between the curves $y=ax^2$ and $x=ay^2$ $(a > 0)$ is $1$ sq. unit. Then the value of $a$ is: