Foundations — Arc length formula — derivation

This page assumes you know nothing. Before you touch the derivation in the parent note, we build every symbol it throws at you — one at a time, each anchored to a picture, each earned before it is used.

0. What a "curve $y=f(x)$" even means

The picture. Draw a horizontal axis (call it $x$, "how far across") and a vertical axis (call it $y$, "how far up"). For every $x$ you slide to, the rule marks one dot at height $y=f(x)$. Join all the dots and you get a curve — a wiggly line living above the $x$-axis positions.

• $a$ and $b$ are the start and end positions on the $x$-axis. We only measure the piece of curve between $x=a$ and $x=b$.

1. The two axes and a point $(x,y)$

The parent writes points like $(x_i,y_i)$ and $(x_{i-1},y_{i-1})$. Read those as:

• $x_i$ ::: the $x$-address of the $i$-th chosen dot.
• The subscript $i$ is just a label counter — "dot number $i$." There is nothing mysterious in it: $x_0$ is the first, $x_1$ the second, and so on up to $x_n$, the last.

2. The gap symbols $\Delta x$ and $\Delta y$

$\Delta x_i=x_i-x_{i-1},\Delta y_i=y_i-y_{i-1}$

The picture. Between two neighbouring dots, draw a small horizontal step and then a small vertical step. The horizontal step is $\Delta x_i$; the vertical step is $\Delta y_i$. Together with the straight line joining the dots, they form a tiny right triangle.

3. Pythagoras — the length of the slanted stick

Apply it to our tiny triangle. The legs are $\Delta x_i$ (across) and $\Delta y_i$ (up); the hypotenuse is the straight chord $L_i$ joining the two dots:

$L_i=\sqrt{(\Delta x_i{)}^2+(\Delta y_i{)}^2}$

4. Slope: the ratio $\frac{\Delta y}{\Delta x}$

The picture. Same tiny triangle. A near-flat chord has a small up-step over a big across-step → small ratio. A near-vertical chord has a big up-step → large ratio.

• This chord-slope, using two separated dots, is called a secant slope.

5. The derivative $\frac{dy}{dx}$ and $f^{\prime }(x)$

The picture. Shrink the tiny triangle toward a point. The chord snuggles right up against the curve and becomes the tangent line — the straight line just kissing the curve there. Its slope is the derivative.

• The little prime mark ${}^{\prime }$ in $f^{\prime }$ just means "the derivative of $f$." The star in $x_i^{\ast }$ means "some special point inside interval $i$" whose existence the theorem guarantees.

6. The summation sign $\sum$

${\sum }_{i=1}^n{L}_i=L_1+L_2+\cdots +L_n$

7. The limit $\lim$ and the integral $\int$

${\lim }_{n\to \infty }{\sum }_{i=1}^n\sqrt{1+(f^{\prime }(x_i^{\ast }){)}^2}\Delta x_i={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx$

• $dx$ at the end of an integral marks "the width of each infinitely-thin slice," the leftover of $\Delta x_i$ after shrinking.

Prerequisite map

Where these tools go next

Once the $ds=\sqrt{d{x}^2+d{y}^2}$ idea is solid here, the same element of arc length powers three neighbouring topics: Surface area of revolution (spin $ds$ around an axis), Parametric arc length (both $x$ and $y$ follow a clock), and Polar arc length (measure with angle and radius instead). Build $ds$ once; reuse it everywhere.

Equipment checklist

Test yourself — say each answer aloud before revealing.