Step by step
The question
Question: find the length of the curve y = 2/3 × x^(3/2) from x = 0 to x = 3.
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 3: the length is this integral.
2 Find f′(x)
The power rule on each term: f′(x) = √x.
x to the one half is √x.
3 Under the root
Square it.
The number is squared, and (√x)² is x: x.
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
4 Integrate
This needs u-substitution.
Let u = 1 + x.
Then du is just dx.
The limits change too: u goes from 1 to 4.
Write it with u.
√u is u to the one half.
Add 1 to the power: three halves.
Divide by three halves: times 2/3.
Multiply the numbers in front: 2/3.
Then top minus bottom.
u to the three halves is u times √u.
So the length is 14/3, about 4.667 units.
5 Check
Let's check with a calculator: it adds up 1000 tiny straight pieces along the curve.
That gives about 4.667, the same as our answer.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.