Step by step
The question
Question: find the surface area when y = 1/3 × x³, from x = 0 to x = 2, turns around the x-axis.
1 The idea
Draw the curve.
Cut the skin into thin bands.
A band goes around a circle of radius f(x): it is 2π f(x) long.
Its width is a tiny piece of the curve, ds, from the arc length lesson.
Not dx: the curve is slanted.
Add up all the bands: ds is the root of 1 plus f′ squared, times dx.
2 Find f′(x)
The power rule: f′(x) = x².
3 Under the root
Square it and add 1: 1 + x⁴.
Careful!
Not dx!
The band is slanted: its width is ds, the root of 1 plus f′ squared, times dx.
4 Integrate
Put f(x) in front of the root.
u-substitution: u = 1 + x⁴.
Its derivative has x³ in it, and so does our integral: a good sign.
u goes from 1 to 17.
1/3 × x³ dx is 1/3 times x³ dx.
x³ dx is du over 4.
So the number in front is π over 6.
√u integrates to two thirds of u to the three halves: π over 9 in front.
Then top minus bottom.
So the surface area is π over 9 times (17√17 − 1) square units.
That is about 24.12.
5 Check
Let's check with a calculator: it adds up 1000 thin bands.
That gives about 24.12, 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.