Step by step
The question
Question: find the length of the curve y = 2/3 × x^(3/2) from x = 0 to x = 8.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The idea
Draw the curve.
This is the piece we want to measure.
The arc length formula: the integral of the root of 1 plus the slope squared, here from 0 to 8.
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.
Add 1.
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
4 Integrate
A root of a sum has no simple rule.
But √u alone is easy.
So call the inside u: u-substitution.
Let u = 1 + x.
Then du is just dx.
The limits change too, because we now count in u: at x = 0, u is 1; at x = 8, u is 9.
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.
9 times √9 is 27, and 1 times √1 is 1.
So the length is 52/3, about 17.333 units.
5 Check
Let's check with a calculator: it adds up 1000 tiny straight pieces along the curve.
That gives about 17.333, the same as our answer.
Right!
6 Pick the choice
Our answer is L = 52/3.
That is choice C.
7 Why not the others
A is 32√2/3.
Wrong: that is the height y(8), not the length.
B is 8.
Wrong: the width only.
D is 40.
Wrong: left out the square root: ∫ (1 + x) dx.
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Then: check.
Then: pick the choice.
Last: why not the others.