Area between y = x^2 - 1 and the x-axis Step by Step

Step by step

The question

Question: find the area between y = x² − 1 and the x-axis.

y = x² − 1

1 Draw the region

Draw the curve first.

2 Where it crosses

Where does it cross the x-axis?

Where y is 0.

x² − 1 = 0

Factor it.

x² − 1 is a difference of two squares: x² minus 1 squared.

So it is (x − 1)(x + 1).

(x + 1)(x − 1) = 0

So x = −1 or x = 1.

x = −1 or x = 1

From −1 to 1 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.

∫ below the x-axis → a minus number

3 The antiderivative

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

F(x) = x³3 − x

4 Each piece

The integral: from −1 to 1.

Top minus bottom.

∫ from −1 to 1 of f(x) dx = F(1) − F(−1)

= (1³3 − 1) − ((−1)³3 − (−1))

= −23 − 23

That is −4/3.

= −43

Careful!

An area is never negative.

The minus only says: below the axis.

Careful: not area = −43 ✗

5 Add the areas

−4/3 is below the axis.

Its area is the size, without the minus: 4/3.

below: area = 43

So the area is 4/3 square units.

area = 43 square units

6 Check

Let's check F: its derivative is x² − 1, the function we started with.

Right!

F′(x) = x² − 1

Let's look back.

First: draw the region.

Then: where it crosses.

Then: the antiderivative.

Then: each piece.

Then: add the areas.

Last: check.

Answer