I A Maron Solutions for Chapter: Applications of the Definite Integral, Exercise 6: Computing the Volume of Solid

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.