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I was searching for an equation to calculate the arc length of the archimedean spiral and found the following notation:

$$ \frac{b}{2}[\theta\sqrt{1+\theta^2}+ln(\theta+\sqrt{1+\theta^2})]^{\theta_2}_{\theta_1} $$

which, by the context, I know it denotes an operation starting in $\theta_1$ and ending in $\theta_2$.

My question is: What does this subscript and superscript mean after the braces? How do I know what to do when I read a notation like this one.

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2 Answers 2

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This is common notation when evaluating integrals.

$$ [f(t)]_{t_0}^{t_1} = f(t_1) - f(t_0), $$ though you usually need to deduce from the context what the variable is, it won’t always be $t$. (It’s $\theta$ in your case.)

Sometimes you will also see $$ [f(t)]_{t=t_0}^{t_1} $$ instead, which explicitly specifies the variable. Another common way of denoting the same thing is $$ f(t) \Big|_{t_0}^{t_1}, $$ again, sometimes with “$t = t_0$” in the subscript to disambiguate the variable.

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Simply put, $[f(x)]_{x_1}^{x_2}=(f(x_2)-f(x_1))$.

This notation comes from calculating a definite integral. Indeed, $$\int_0^1 x\mathrm{d}x=\left[\frac{x^{2}}{2}\right]_0^{1}=\frac{1}{2}-0=\frac{1}{2}.$$

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  • $\begingroup$ I'm slow using this new keyboard. Enjoy the two very similar answers! $\endgroup$ Nov 18, 2021 at 13:33

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