Solid of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane.

Formulas for solids of revolution

Rotations about the y-axis

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x=a and x=b about the y-axis is given by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:V = 2pi int_a^b x,dx

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If one of the bounding curves is actually the x-axis, then we can let g(x) = 0 in the formula above, and we have:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:V = 2pi int_a^b x f(x),dx

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Rotations about the x-axis

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x=a and x=b about the x-axis is given by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:V = pi int_a^b f(x)^2 - g(x)^2,dx

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

As above, we can use

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:V = pi int_a^b f(x)^2,dx

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

if one of the bounding curves is actually the x-axis.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~