Step by step
The question
Question: find the volume when y = −x² + 4x from x = 0 to x = 4 turns around the x-axis.
We turn a flat region around the x-axis, and slice the solid it makes.
1 Draw and turn it
Draw the curve first.
The region under the curve, from x = 0 to x = 4, 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 = −x² + 4x.
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 4: 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.
The right way: V = π∫ from 0 to 4 of (−x² + 4x)² dx.
4 Square it
Square the radius.
A bracket squared is the bracket times itself: multiply every part by every part.
Put it in the integral.
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
6 Top minus bottom
Put in 4, then take away the same with 0.
So the volume is 512π over 15 cubic units.
That is about 107.23.
7 Check
Let's check with a calculator: it adds up 1000 thin disks.
That gives about 107.23, the same as our answer.
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.