My Favorite One-Liners: Part 25

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Consider the integral

\displaystyle \int_0^2 2x(1-x^2)^3 \, dx

The standard technique — other than multiplying it out — is using the substitution u = 1-x^2. With this substitution du = -2x \, dx. Also, x = 0 corresponds to u = 1, while x = 2 corresponds to u = -3. Therefore,

\displaystyle\int_0^2 2x(1-x^2)^3 \, dx = - \displaystyle\int_0^2 (-2x)(1-x^2)^3 \, dx = -\displaystyle\int_1^{-3} u^3 \, du.

My one-liner at this point is telling my students, “At this point, about 10,000 volts of electricity should be going down your spine.” I’ll use this line when a very unexpected result happens — like a “left” endpoint that’s greater than the “right” endpoint. Naturally, for this problem, the next step — though not logically necessary, it’s psychologically reassuring — is to absorb the negative sign by flipping the endpoints:

\displaystyle\int_0^2 2x(1-x^2)^3 \, dx =  -\displaystyle\int_1^{-3} u^3 \, du = \displaystyle\int_{-3}^1 u^3 \, du,

and then the calculation can continue.

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