Shell integration

Definition

The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be:

${\displaystyle 2\pi {\int }_a^{b}xf(x)dx}$

If the function is of the y coordinate and the axis of rotation is the x-axis then the formula becomes:

${\displaystyle 2\pi {\int }_a^{b}yf(y)dy}$

If the function is rotating around the line x = h then the formula becomes:

${\displaystyle \{\begin{array}{ll}{\displaystyle 2\pi {\int }_a^b(x-h)f(x)dx,}& \text{if}h\le a<b\\ {\displaystyle 2\pi {\int }_a^b(h-x)f(x)dx,}& \text{if}a<b\le h,\end{array}}$

and for rotations around y = k it becomes

${\displaystyle \{\begin{array}{ll}{\displaystyle 2\pi {\int }_a^b(y-k)f(y)dy,}& \text{if}k\le a<b\\ {\displaystyle 2\pi {\int }_a^b(k-y)f(y)dy,}& \text{if}a<b\le k.\end{array}}$

The formula is derived by computing the double integral in polar coordinates.

Derivation of the formula