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

Step by step

The question

Question: find the area between y = x² − x − 2 and the x-axis.

y = x² − x − 2

1 Draw the region

Draw the curve first.

2 Where it crosses

Where does it cross the x-axis?

Where y is 0.

x² − x − 2 = 0

Factor it.

We need two numbers that multiply to −2 and add to −1: 1 and −2.

(x + 1)(x − 2) = 0

So x = −1 or x = 2.

x = −1 or x = 2

From −1 to 2 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 − x²2 − 2x

4 Each piece

The integral: from −1 to 2.

Top minus bottom.

∫ from −1 to 2 of f(x) dx = F(2) − F(−1)

= (2³3 − 2²2 − 2(2)) − ((−1)³3 − (−1)²2 − 2(−1))

= −103 − 76

That is −9/2.

= −92

Careful!

An area is never negative.

The minus only says: below the axis.

Careful: not area = −92 ✗

5 Add the areas

−9/2 is below the axis.

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

below: area = 92

So the area is 9/2 square units.

area = 92 square units

6 Check

Let's check F: its derivative is x² − x − 2, the function we started with.

Right!

F′(x) = x² − x − 2

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