Compute the volume of the solid torus. The torus is a solid generated by revolving a circle of radius $a$ about an axis lying in its plane at a distance $b$ from the centre $\left(b\ge a\right)$. (A tire, for example, has the form of the torus).

Compute the volume of the solid generated by revolving the figure bounded by the following curve and lines :

$y=2x-x^2,y=0$ about the $x$-axis;

Compute the volume of the solid generated by revolving the figure $y=x^3$ bounded by the lines $y=0,x=2$ about the $y$-axis.

Compute the volume of the solid generated by revolving the figure $y=\sin x$ (one wave) bounded by $y=0$ about the $x$-axis.

Find the areas of the figures bounded by the curves represented parametrically :

$x=2t-t^2; y=2t^2-t^3$

Find the areas of the figures bounded by the curves represented parametrically :

$x=\frac{t^2}{1+t^2}; y=\frac{t\left(1-t^2\right)}{1+t^2}$

Find the area of the figure bounded by the curve given in polar coordinates as $\rho =a \sin 3\phi$ (a three-leaved rose).

Find the areas of the figures bounded by the curves given in polar coordinates :

$\rho =3\sin \phi$ and $\rho =\sqrt{3}\cos \phi$