The length of a curve

What you will be able to do

• write the length of a curve as an integral and check it on a straight line;
• find the exact length when $1 + [f'(x)]^2$ is a perfect square;
• estimate a length by adding the lengths of many tiny straight pieces.

Where this sits. This is an enrichment lesson. The Indian syllabus (NCERT Class 12 and JEE Main) does not include the length of a curve. AP Calculus BC does. Other syllabuses differ, so check your own syllabus before you spend time on the exact examples. The idea, adding thin pieces, is the same as in the area and volume lessons.

Adding tiny straight pieces

How long is a hanging cable, or a winding path? A ruler does not bend, but a very short piece of a smooth curve is almost straight. Over a tiny step, the curve moves $dx$ along and $dy$ up. By the theorem of Pythagoras, the length of the piece is

Take $dx$ out of the root: $ds = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}\,dx$. Add the pieces from $x = a$ to $x = b$:

A check with a straight line

Take the line $y = \dfrac{3}{4}x$ from $x = 0$ to $x = 8$. Its slope is $\dfrac{3}{4}$, so $\sqrt{1 + \dfrac{9}{16}} = \dfrac{5}{4}$ and $L = \dfrac{5}{4} \times 8 = 10$. A right triangle with legs $8$ and $6$ has hypotenuse $10$, so the formula agrees with Pythagoras.

Why the exact answers are rare

The square root makes most integrals hard. A parabola, for example, gives a length with a logarithm in it. The exact examples that appear in books are chosen so that $1 + [f'(x)]^2$ is a perfect square. Here are two.

First example. Let $y = \dfrac{2}{3}x^{3/2}$. Then $f'(x) = x^{1/2}$ and $1 + [f'(x)]^2 = 1 + x$. From $x = 0$ to $x = 3$,

Second example. Let $y = \cosh x$. Then $f'(x) = \sinh x$ and $1 + \sinh^2 x = \cosh^2 x$. So $\sqrt{1 + [f'(x)]^2} = \cosh x$, and from $x = 0$ to $x = 1$,

The graph of $\cosh$ is the shape of a hanging chain, called a catenary. A chain hangs as $y = a\cosh\dfrac{x}{a}$, where $a$ depends on the tension and the weight.

Many tiny pieces on the computer

When no neat answer exists, add up the pieces by computer. Choose $n + 1$ points on the curve, join them with straight lines, and add the lengths. As $n$ grows, the sum gets closer to the true length.

A worked example

Find the length of $y = \dfrac{x^3}{6} + \dfrac{1}{2x}$ from $x = 1$ to $x = 2$.

1. Differentiate. $f'(x) = \dfrac{x^2}{2} - \dfrac{1}{2x^2}$.
2. Look for a square. $1 + [f'(x)]^2 = 1 + \dfrac{x^4}{4} - \dfrac{1}{2} + \dfrac{1}{4x^4} = \left(\dfrac{x^2}{2} + \dfrac{1}{2x^2}\right)^2$.
3. Integrate. $L = \displaystyle\int_1^2 \left(\dfrac{x^2}{2} + \dfrac{1}{2x^2}\right)dx = \left[\dfrac{x^3}{6} - \dfrac{1}{2x}\right]_1^2 = \left(\dfrac{4}{3} - \dfrac{1}{4}\right) - \left(\dfrac{1}{6} - \dfrac{1}{2}\right) = \dfrac{17}{12}$.

Practise with new numbers

Try it in Python

The program joins $n + 1$ points of $y = \cosh x$ on $[0, 1]$ by straight pieces and prints the total length. Then it prints the exact value $\sinh 1$.

Common mistakes

• Leaving out the square root. The length element is $\sqrt{1 + [f'(x)]^2}\,dx$, not $(1 + [f'(x)]^2)\,dx$.
• Leaving out the $1$. The $1$ is the $dx^2$ part of $ds^2 = dx^2 + dy^2$.
• Using the chord. The straight line between the end points is shorter than the curve, unless the curve is a straight line.
• Squaring after integrating. Square $f'(x)$ first, add $1$, then take the root, and only then integrate.
• Forcing an exact answer. Most curves have no neat length. A numerical value is a fine answer.

Remember these

• $L = \displaystyle\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$, from $ds^2 = dx^2 + dy^2$.
• For a straight line the formula gives the length by Pythagoras.
• Neat exact answers come from curves where $1 + [f'(x)]^2$ is a perfect square.
• The sum of many tiny chords is always a little below the true length.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the length of the line $y = 2x + 1$ from $x = 0$ to $x = 3$.

2. Find the length of $y = \dfrac{2}{3}x^{3/2}$ from $x = 0$ to $x = 15$.

3. A student writes $L = \displaystyle\int_a^b \left(1 + f'(x)\right)dx$. Spot the mistake.

4. Find the length of $y = \ln(\cos x)$ from $x = 0$ to $x = \dfrac{\pi}{3}$.

5. Is the length of a curve between two points always at least the distance in a straight line between them?

References

• OpenStax, Calculus Volume 1, Chapter 6, section 6.4 (arc length of a curve and surface area).