The area bounded by the curve $y = \sin \left(\frac{x}{3}\right)$,the $x$-axis,and the lines $x = 0$ and $x = 3\pi$ is

Prove that the curves $y^{2}=4x$ and $x^{2}=4y$ divide the area of the square bounded by $x=0, x=4, y=4$ and $y=0$ into three equal parts.

The area enclosed between the curve $y = \log_e(x + e)$ and the coordinate axes is

If the tangent to the curve $y = 1 - x^2$ at $x = \alpha,$ where $0 < \alpha < 1,$ meets the axes at $P$ and $Q.$ Also $\alpha$ varies,the minimum value of the area of the triangle $OPQ$ is $k$ times the area bounded by the axes and the part of the curve for which $0 < x < 1,$ then $k$ is equal to

The area of the region bounded by the curve $y = 2 - x - 3x^2$,the $X$-axis,the $Y$-axis,and the line $x = -2$ is

The area of the region bounded by the $x$-axis and the curves defined by $y = \tan x$ for $(-\pi/3 \le x \le \pi/3)$ is