The length of a curve

The length of a curve is also found with a definite integral¹. We compute the length of $y = \dfrac{2}{3}x^{3/2}$ over $0 \leq x \leq 3$ .

It starts from the Pythagorean theorem

Replacing the curve by a fine polyline, one segment has length $\sqrt{\Delta x^2 + \Delta y^2}$ . Factoring out $\Delta x$ turns it into $\sqrt{1 + \left( \dfrac{\Delta y}{\Delta x} \right)^2}\,\Delta x$ , and summing and taking the limit gives the formula.

L = \int_a^b \sqrt{1 + f'(x)^2}\,dx

The formula for length is the formula for the distance between two points, chopped fine and added up.

Working it out

For this function $f'(x) = x^{1/2}$ , so $1 + f'(x)^2 = 1 + x$ and the inside of the root comes out clean.

L = \int_0^3 \sqrt{1 + x}\,dx = \left[ \frac{2}{3}(1 + x)^{3/2} \right]_0^3 = \frac{2}{3}(8 - 1) = \frac{14}{3}

Quantity	Value
Length of the curve	$\dfrac{14}{3} \approx 4.667$
The segment joining the ends	$\sqrt{21} \approx 4.583$

The curve is a little longer than the segment, which also confirms that the shortest path between two points is a straight line.

Checking against a line

For $y = x$ we have $f' = 1$ , so $L = \int_0^a \sqrt{2}\,dx = \sqrt{2}\,a$ , matching the distance from the origin to $(a, a)$ . The formula returns the obvious answer properly.

When the computation turns hard

Curve	Integral for the length	Result
$y = \dfrac{2}{3}x^{3/2}$	$\int\sqrt{1+x}\,dx$	elementary
$y = x^2$	$\int\sqrt{1+4x^2}\,dx$	involves a logarithm
An ellipse		not expressible in elementary functions²

When the inside of the root is not a square, the computation becomes hard. An integral for a length is a far tougher opponent than one for an area.

The circumference of a circle

For $y = \sqrt{r^2 - x^2}$ we have $f' = -\dfrac{x}{\sqrt{r^2 - x^2}}$ and $\sqrt{1 + f'^2} = \dfrac{r}{\sqrt{r^2 - x^2}}$ . Integrating that from $-r$ to $r$ gives $\pi r$ , the correct length of a semicircle.

The curve rising to the upper right is $y = \dfrac{2}{3}x^{3/2}$ , the curve above it is the integrand $y = \sqrt{1 + x}$ , and the large dots are the two ends.

Arc length, Wikipedia
Elliptic integral, Wikipedia