Volume of y = x turned around the x-axis from x = 0 to x = 3

Step by step

The question

Question: find the volume when y = x 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 = x.

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: x².

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 9π cubic units.

That is about 28.27.

7 Check

Let's check: the solid is a cone, radius 3 and height 3.

A cone's volume is one third π r² h: 9π.

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.