Step by step
The question
Question: find the surface area when y = x, 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)
A line: its slope is the number in front of x, 1.
3 Under the root
Square it and add 1: 2.
Careful!
Not dx!
The band is slanted: its width is ds, the root of 1 plus f′ squared, times dx.
4 Integrate
The root is a number here: it waits in front with 2π.
The antiderivative: add 1 to each power, divide by the new power.
Top minus bottom: 2.
So the surface area is 4√2 π square units.
That is about 17.77.
5 Check
Let's check: the skin is a cone with radius 2 and slanted side 2.828 (Pythagoras).
A cone's side area is π r ℓ: about 17.77.
The same.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.