The area of the region bounded by the curves $y = \sin x$, $y = \cos x$ and the lines $x = 0$, $x = \frac{\pi}{4}$ is

If $\int\limits_0^1 {(4x^3 - f(x))f(x)dx = \frac{4}{7}}$,then the area of the region bounded by $y = f(x)$,the $x$-axis,and the ordinates $x = 1$ and $x = 2$ is:

The area of the region bounded by the curve $y = 2x - x^2$ and the $X$-axis is . . . . . . sq. units.

If $A$ is the area of the region bounded by the curve $y = \sqrt{3x + 4}$,the $x$-axis,and the lines $x = -1$ and $x = 4$,and $B$ is the area bounded by the curve $y^2 = 3x + 4$,the $x$-axis,and the lines $x = -1$ and $x = 4$,then $A:B$ is equal to:

The area of the region bounded by the curve $y = 2^{kx}$ and the lines $x = 0$ and $x = 2$ in the first quadrant is $\frac{2}{\ln 2}$. Find the value of $k$.

The straight line through the origin which divides the area formed by the curves $y=2x-x^2$, $y=0$, and $x=1$ into two equal halves is