Step by step
The question
Question: find the volume when the region between y = x and y = x² turns around the x-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 = 1.
The region between them, from x = 0 to x = 1, is shaded.
3 One slice
Cut the solid into thin slices.
Each slice is a washer: a flat disk with a hole.
The outside radius is the curve further from the axis: R = x.
The inside radius, the hole, is the other one: r = x².
A washer is the big circle minus the hole: π R² minus π r², times its thickness dx.
Which curve is outside?
Test a number between: x is bigger there.
4 Write the integral
Add up all the slices from 0 to 1: that is an integral.
π is just a number, so it waits in front.
Careful!
Not R minus r, squared!
The hole is a whole circle: take away π r squared.
5 Square it
Square both radii, then take away.
Square the number and the power: x².
And the hole: x⁴.
6 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
7 Top minus bottom
Put in 1, then take away the same with 0.
So the volume is 2π over 15 cubic units.
That is about 0.42.
8 Check
Let's check with a calculator: it adds up 1000 thin washers.
That gives about 0.42, the same as our answer.
Right!
Let's look back.
First: draw and turn it.
Then: where they meet.
Then: one slice.
Then: write the integral.
Then: square it.
Then: the antiderivative.
Then: top minus bottom.
Last: check.