Last time, we talked about how the classical $\mathscr{L}^2$ norm of a (one-variable) function could be visualized as a solid consisting of squares. Of course, there's another formula that is defined by almost the same formula, with an extra factor of $\pi$: the

*solid of revolution*of the function about the $x$-axis: \[

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

\]

It looks pretty cool, although not quite as cool as twisted square cross-sections.

I wanted to mention another common application of $\mathscr{L}^2$ norms that probably is more familiar than any other: the statistical concept of

*variance*. For functions $f$ defined on a continuous domain, we can define an average value:\[

\overline{f} = \frac1{b-a}\int_a^b f(x)\,dx.

\] which basically is the height of a rectangle over the same domain $[a,b]$

*that has the same area as the area under*$f$. For example, for \[

f(x) = \sin \left( x \right)\sin \left( t \right)+\frac{1}{3}\sin \left( 5x \right)\sin \left( 5t \right)+\frac{1}{3}\sin \left( 10x \right)\sin \left( 10t \right)+\frac{1}{5}\sin \left( 15x \right)\sin \left( 15t \right)

\]

(choosing $t=3.030303$), it looks like:

*which is precisely the same quantity as the solid obtained by revolving $f(x)$ around its mean $y=\overline{f}$.*The resulting solid is the title picture.

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