The area bounded by the curve $y = f(x)$,the coordinate axes,and the line $x = x_1$ is given by $x_1 \cdot e^{x_1}$. Therefore,$f(x)$ equals:

The area enclosed by the curve $y^2 + x^4 = x^2$ is :

The area of the region enclosed by the curves $y=e^x$,$y=|e^x-1|$ and the $y$-axis is:

If a straight line $y - x = 2$ divides the region $x^2 + y^2 \le 4$ into two parts,then the ratio of the area of the smaller part to the area of the greater part is

The area bounded by the curve $y = x |\sin(\frac{x}{2})|$ and the $X$-axis between the lines $x = 0$ and $x = 4\pi$ (in square units) is: (in $\pi$)

Let $S(\alpha) = \{(x,y) : y^2 \leq x, 0 \leq x \leq \alpha\}$ and $A(\alpha)$ be the area of the region $S(\alpha)$. If for a $\lambda, 0 < \lambda < 4, A(\lambda) : A(4) = 2 : 5$,then $\lambda$ equals: