Find the area under the curve $y=x^{4}$ bounded by the lines $x=1$,$x=5$ and the $x$-axis.

The area bounded by the curve $y = x^2 + 2$,the $x$-axis,and the lines $x = 1$ and $x = 2$ is:

The area bounded by the curve $y = x^2 + 4x + 5$,the coordinate axes,and the minimum ordinate is:

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:

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

The area of the curve $xy^2 = a^2(a - x)$ bounded by the $y$-axis is