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

$A$ curve satisfying the initial condition $y(1)= 0$ satisfies the differential equation $x \frac{dy}{dx}= y -x^2$. The area bounded by the curve and the $x$-axis is:

Find the area bounded by the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the ordinates $x=0$ and $x=ae$,where $b^{2}=a^{2}(1-e^{2})$ and $e < 1$.

The area of the region bounded by the curve $y=|x-5|$,the $X$-axis,and the lines $x=5$ and $x=6$ is . . . . . . sq. units.

The area (in sq. units) bounded by the parabola $y=x^2+3$,the tangent to the parabola at $(3,12)$ and the coordinate axes and lying in the first quadrant is

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