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:
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