Area between y = x^2 - 3x and the x-axis

Step by step

The question

Question: find the area between y = x² − 3x and the x-axis.

1 Draw the region

Draw the curve first.

2 Where it crosses

Where does it cross the x-axis?

Where y is 0.

Factor it.

We need two numbers that multiply to 0 and add to −3: 0 and −3.

So x = 0 or x = 3.

From 0 to 3 it is below the axis: shaded red.

An integral counts area below the x-axis as negative.

But an area is never negative.

So we work out each piece on its own.

3 The antiderivative

The antiderivative: add 1 to each power, divide by the new power.

Now we have F(x) = x³ over 3 − 3x² over 2.

4 Each piece

The integral: from 0 to 3.

Top minus bottom.

That is −9/2.

Careful!

An area is never negative.

The minus only says: below the axis.

5 Add the areas

−9/2 is below the axis.

Its area is the size, without the minus: 9/2.

So the area is 9/2 square units.

6 Check

Let's check F: its derivative is x² − 3x, the function we started with.

Right!

Let's look back.

First: draw the region.

Then: where it crosses.

Then: the antiderivative.

Then: each piece.

Then: add the areas.

Last: check.