# Tag Info

Accepted

### Graphs for which a calculus student can reasonably compute the arclength

Ferdinands, in his short note "Finding Curves with Computable Arc Length", also comments on the difficulty of coming up with suitable examples of curves with easily-computable arclengths. In ...
Accepted

### Is the arc length always irrational between two rational points?

Obviously, a straight line between two rational points can have rational length $-$ just take $(0,0)$ and $(1,0)$ as your rational points. But a curved line can also have rational length. Consider ...
• 65k

### Is the arc length always irrational between two rational points?

An example of a curve with rational arc lengths between at least some pairs of rational points is a cardioid. Down to scaling and rotation, a cardioid may be rendered in polar coordinates by the ...
• 40.6k

### Graphs for which a calculus student can reasonably compute the arclength

I will dissent here (as often) and say: DON'T. The problem here is looking at everything as something that needs to be computed, "solved", or otherwise manipulated into some set, pat form. ...
• 19.8k

### Graphs for which a calculus student can reasonably compute the arclength

Another example: you can get $$\sqrt{1 + [f'(x)]^2} = ax + \frac 1{2ax}$$ by taking $f(x) = \frac 12 a x^2 - \frac 1{4a} \ln(x)$ for any constant $a$. A possibly helpful way of reframing the ...
• 227k

### Is the arc length always irrational between two rational points?

So, my question is that do all curved path have irrational lengths? Of course not. A circle with radius $\frac{1}{2\pi}$ is a curved path and has length $1$ which is a rational number. If you put the ...
• 124k

### Graphs for which a calculus student can reasonably compute the arclength

This example $$y = a\cosh \frac{x}{a}$$ is quite simple for computations.
• 3,666
Accepted

### Conjectured connection between $e$ and $\pi$ in a semidisk.

$\newcommand{\eps}{\varepsilon}$ The answer is in fact not $\pi/2$ but $$\frac{3}{(2\pi)^{1/3}}\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 1.56813\ldots.$$ This product ...
• 33.7k