The area of the region bounded by the line $y=2x+1$,the $X$-axis,and the ordinates $x=-1$ and $x=1$ is

The area of the region bounded by the curve $y = x^3$,and the lines $y = 8$ and $x = 0$ is

The area of the region bounded by the curves $y = \sin x$,$y = \cos x$ and $x = 0$ 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:

The area bounded by the curve $4y^{2} = x^{2}(4-x)(x-2)$ is equal to ...... .

The area (in sq unit) of the region bounded by the curves $2x = y^2 - 1$ and $x = 0$ is