Step by step
The question
Question: find the volume when y = 2x from x = 0 to x = 3 turns around the x-axis.
1 Draw and turn it
Draw the curve first.
The region under the curve, from x = 0 to x = 3, is shaded.
2 One slice
Cut the solid into thin slices.
Each slice is a disk: a very flat cylinder.
Its radius is the height of the curve: r = 2x.
Its thickness is a tiny dx.
A cylinder is its circle's area times its thickness.
So one slice is π r² dx.
3 Write the integral
Add up all the slices from 0 to 3: that is an integral.
π is just a number, so it waits in front.
Careful!
Square the radius!
A circle's area is π r squared, not π r.
4 Square it
Square the radius.
Square the number and the power: 4x².
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
6 Top minus bottom
Put in 3, then take away the same with 0.
So the volume is 36π cubic units.
That is about 113.10.
7 Check
Let's check: the solid is a cone, radius 6 and height 3.
A cone's volume is one third π r² h: 36π.
The same.
Right!
Let's look back.
First: draw and turn it.
Then: one slice.
Then: write the integral.
Then: square it.
Then: the antiderivative.
Then: top minus bottom.
Last: check.