Step by step
The question
Question: find the length of the curve y = 1/8 × x⁴ + 1/(4x²) from x = 1 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 1 to 2: the length is this integral.
2 Find f′(x)
The power rule on each term: f′(x) = 1/2 × x³ − 1/(2x³).
3 Under the root
Square the bracket: the first squared, plus 2 times the first times the second, plus the second squared.
The middle term is −1/2.
Add 1: the middle term becomes + 1/2.
Look: this is a square too!
The same terms with a plus in the middle.
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
4 Integrate
The root of a square is the bracket itself.
For x from 1 to 2 it is positive.
The antiderivative: add 1 to each power, divide by the new power.
5 Top minus bottom
Put in 2, then take away the same with 1.
So the length is 33/16, about 2.063 units.
6 Check
Let's check with a calculator: it adds up 1000 tiny straight pieces along the curve.
That gives about 2.062, the same as our answer.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Then: top minus bottom.
Last: check.