Step by step
The question
Question: find the volume when the region between y = 2x and y = x² turns around the y-axis.
1 Draw and turn it
Draw the curves first.
2 Where they meet
The region ends where the curves meet: where the two y's are equal.
Move everything to one side.
Times −1 on both sides: every sign changes, and 0 stays 0.
Now x² comes first with a plus.
Every term has an x in it: take it out.
A product is 0 when one of its parts is 0.
So x = 0, or x = 2.
The region between them is shaded.
3 One shell
Cut the solid into thin tubes, one inside the other.
Each one is a shell: like a can with no top and no bottom.
A shell at x is x away from the y-axis.
That is its radius.
Its height is the top curve minus the bottom one.
Which is on top?
Test a number between: 2x is bigger there.
So h = 2x − x².
Cut a shell open and lay it flat.
It is a thin slab: 2πx long, the distance around a circle of radius x.
h tall, and dx thick.
4 Write the integral
Add up all the shells from 0 to 2: that is an integral.
2π is just a number, so it waits in front.
Careful!
The radius is not the height of the curve.
It is the distance to the y-axis: x.
5 Multiply
Multiply x into the bracket: x times 2x − x².
x times every part.
6 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
7 Top minus bottom
Put in 2, then take away the same with 0.
2π times 4/3: the volume is 8π over 3 cubic units.
That is about 8.38.
8 Check
Let's check with a calculator: it adds up 1000 thin shells.
That gives about 8.38, the same as our answer.
Right!
Let's look back.
First: draw and turn it.
Then: where they meet.
Then: one shell.
Then: write the integral.
Then: multiply.
Then: the antiderivative.
Then: top minus bottom.
Last: check.