Step by step
The question
Question: find the surface area when y = √x, from x = 1 to x = 4, 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)
√x is x to the one half.
The power rule gives 1/2√x.
3 Under the root
Square it: 1/4x.
Over one bottom: 1 is 4x/4x.
Careful!
Not dx!
The band is slanted: its width is ds, the root of 1 plus f′ squared, times dx.
4 Simplify
Now f(x) times the root.
The root of a fraction is the root of the top over the root of the bottom.
The root of 4x is 2√x.
√x and √x cancel: only the root of 4x + 1, over 2, is left.
5 Integrate
2π times one half is π: it waits in front.
u-substitution: u = 4x + 1, so dx is du over 4.
u goes from 5 to 17.
With u: the 1/4 comes out.
√u integrates to two thirds of u to the three halves.
Multiply the numbers in front: π over 6.
Then top minus bottom.
So the surface area is π over 6 times (17√17 − 5√5) square units.
That is about 30.85.
6 Check
Let's check with a calculator: it adds up 1000 thin bands.
That gives about 30.85, the same as our answer.
Right!
Let's look back.
First: the idea.
Then: find f′(x).
Then: under the root.
Then: simplify.
Then: integrate.
Last: check.