Worked examples
The question
Question: find the length of the curve y = 3/4 × x from x = 0 to x = 4.
y = 34x
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.
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).
ds = √1 + (dydx)2 dx
Add up all the tiny pieces from 0 to 4: the length is this integral.
L = ∫ from 0 to 4 of √1 + (f′(x))2 dx
2 Find f′(x)
The power rule on each term: f′(x) = 3/4.
f′(x) = 34
3 Under the root
Square the slope: 9/16.
(f′(x))2 = (34)2 = 916
Add 1: 25/16.
A straight line has the same slope all along.
1 + (f′(x))2 = 1 + 916 = 2516
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
Careful: not √1 + f′(x) ✗
4 Integrate
L = ∫ from 0 to 4 of √2516 dx
The root of 25/16 is 5/4.
L = ∫ from 0 to 4 of 54 dx
√2516 = 54
A number's integral is the number times the width, 4 − 0.
= 54 × (4 − 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 4 across and 3 up.
Pythagoras gives 5.000.
The same.
Right!
distance: √4² + 3² = √25 ≈ 5.000
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.
▶ Watch it solved
The question
Question: find the length of the curve y = x^(3/2) from x = 0 to x = 4.
y = x^(3/2)
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.
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).
ds = √1 + (dydx)2 dx
Add up all the tiny pieces from 0 to 4: the length is this integral.
L = ∫ from 0 to 4 of √1 + (f′(x))2 dx
2 Find f′(x)
The power rule on each term: f′(x) = 3/2 × √x.
x to the one half is √x.
f′(x) = 32√x
3 Under the root
Square it.
The number is squared, and (√x)² is x: 9/4 × x.
(f′(x))2 = (32√x)2 = 94x
1 + (f′(x))2 = 1 + 94x
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
Careful: not √1 + f′(x) ✗
4 Integrate
This needs u-substitution.
L = ∫ from 0 to 4 of √1 + 94x dx
Let u = 1 + 9/4 × x.
Then du is 9/4 dx, so dx is 4/9 du.
The limits change too: u goes from 1 to 10.
u = 1 + 94x
du = 94 dx → dx = 49 du
x = 0 → u = 1, x = 4 → u = 10
Write it with u.
The 4/9 waits in front.
L = 49∫ from 1 to 10 of √u du
√u is u to the one half.
Add 1 to the power: three halves.
Divide by three halves: times 2/3.
= 49 × 23[ u^(3/2) ] from 1 to 10
Multiply the numbers in front: 8/27.
Then top minus bottom.
= 827(10^(3/2) − 1^(3/2))
u to the three halves is u times √u.
u^(3/2) = u√u
So the length is 8/27 times (10√10 − 1), about 9.073 units.
L = 827(10√10 − 1) ≈ 9.073
5 Check
Let's check with a calculator: it adds up 1000 tiny straight pieces along the curve.
That gives about 9.073, the same as our answer.
Right!
1000 thin straight pieces: ≈ 9.073
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.
Answer L = 827(10√10 − 1) ≈ 9.073
▶ Watch it solved
The question
Question: find the length of the curve y = 1/6 × x³ + 1/(2x) from x = 1 to x = 2.
y = 16x3 + 12x
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.
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).
ds = √1 + (dydx)2 dx
Add up all the tiny pieces from 1 to 2: the length is this integral.
L = ∫ from 1 to 2 of √1 + (f′(x))2 dx
2 Find f′(x)
The power rule on each term: f′(x) = 1/2 × x² − 1/(2x²).
f′(x) = 12x2 − 12x2
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.
(f′(x))2 = 14x4 − 12 + 14x4
2 × 12 × (−12) = −12
Add 1: the middle term becomes + 1/2.
1 + (f′(x))2 = 14x4 + 12 + 14x4
Look: this is a square too!
The same terms with a plus in the middle.
= (12x2 + 12x2)2
(a + b)² = a² + 2ab + b²
Careful!
The slope must be squared first: it is the root of 1 plus f′ squared.
Careful: not √1 + f′(x) ✗
4 Integrate
The root of a square is the bracket itself.
For x from 1 to 2 it is positive.
L = ∫ from 1 to 2 of (12x2 + 12x2) dx
The antiderivative: add 1 to each power, divide by the new power.
L = [ x36 − 12x ] from 1 to 2
5 Top minus bottom
Put in 2, then take away the same with 1.
= (1 × 236 − 12 × 21) − (1 × 136 − 12 × 11)
= 1312 − −13
So the length is 17/12, about 1.417 units.
L = 1712 ≈ 1.417
6 Check
Let's check with a calculator: it adds up 1000 tiny straight pieces along the curve.
That gives about 1.417, the same as our answer.
Right!
1000 thin straight pieces: ≈ 1.417
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Then: top minus bottom.
Last: check.
▶ Watch it solved
