Step by step
The question
Question: find the length of the curve y = x from x = 0 to x = 2.
1 The idea
Draw the curve.
This is the piece we want to measure.
Zoom in on a tiny piece.
It is almost straight: a slanted side ds, over dx and up dy.
Pythagoras on the tiny triangle: ds² = dx² + dy².
Take dx² out of the root: ds is the root of 1 plus dy over dx, squared, times dx.
dy over dx is the slope, f′(x).
Add up all the tiny pieces from 0 to 2: the length is this integral.
2 Find f′(x)
The power rule on each term: f′(x) = 1.
3 Under the root
Square the slope: 1.
Add 1: 2.
A straight line has the same slope all along.
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
4 Integrate
Simplify the root.
A number's integral is the number times the width, 2 − 0.
So the length is 2√2, about 2.828 units.
5 Check
Let's check: a straight line's length is the distance between its ends.
It goes 2 across and 2 up.
Pythagoras gives 2.828.
The same.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.