Step by step
The question
Question: find the surface area when y = √(16 − x²), 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 chain rule: the root's derivative is 1/2 roots, times the inside's slope, −2x.
The 2s cancel.
3 Under the root
Square it: x² over 16 − x².
Over one bottom: the x²'s cancel on top, 16 is left.
Careful!
Not dx!
The band is slanted: its width is ds, the root of 1 plus f′ squared, times dx.
4 Integrate
f(x) times the root: the root of 16 is 4, and the root of 16 − x² cancels.
Only 2π times 4 is left: a number!
A number's integral is the number times the width.
So the surface area is 16π square units.
That is about 50.27.
5 Check
Let's check: a band of a sphere has area 2π r times its width h, here 2.
That is 16π.
The same.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: integrate.
Last: check.