What is the length of the curve y = (4/3)x from x = 0 to x = 3? Step by Step

Step by step

The question

Question: find the length of the curve y = 4/3 × x from x = 0 to x = 3.

y = 43x

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 3.

L = ∫ from 0 to 3 of √1 + (f′(x))² dx

2 Find f′(x)

The power rule on each term: f′(x) = 4/3.

f′(x) = 43

3 Under the root

Square the slope: 16/9.

(f′(x))² = (43)² = 169

Add 1: 25/9.

A straight line has the same slope all along.

1 + (f′(x))² = 1 + 169 = 259

Careful!

The slope must be squared first: it is the root of 1 plus f′ squared.

Careful: not √1 + f′(x) ✗

4 Integrate

Put it under the root.

L = ∫ from 0 to 3 of √259 dx

The root of 25/9 is 5/3.

√259 = 53

So the integral is this.

L = ∫ from 0 to 3 of 53 dx

A number's integral is the number times the width, 3 − 0.

= 53 × (3 − 0)

So the length is 5 units.

L = 5

5 Check

Let's check: a straight line's length is the distance between its ends.

It goes 3 across and 4 up.

Pythagoras gives 5.000.

The same.

Right!

distance: √3² + 4² = √25 ≈ 5.000

6 Pick the choice

Our answer is L = 5.

That is choice A.

A) 5

7 Why not the others

B is 3.

Wrong: the width only.

B) 3 ✗ the width only

C is 4.

Wrong: the height only.

C) 4 ✗ the height only

D is 7.

Wrong: added 3 and 4: the length is √(3² + 4²).

D) 7 ✗ added 3 and 4: the length is √(3^2 + 4^2)

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.

Answer