Area between y = x - 2 and the x-axis from x = 0 to x = 4 Step by Step

Step by step

The question

Question: find the area between y = x − 2 and the x-axis from x = 0 to x = 4.

y = 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.

x − 2 = 0

Factor it.

(x − 2) = 0

So x = 2.

x = 2

From 0 to 2 it is below the axis: shaded red.

From 2 to 4 the curve is above the axis: shaded green.

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) = x²2 − 2x

4 Each piece

The first piece: from 0 to 2.

Top minus bottom.

∫ from 0 to 2 of f(x) dx = F(2) − F(0)

= (2²2 − 2(2)) − (0²2 − 2(0))

= −2 − 0

That is −2.

= −2

Now the second piece: from 2 to 4.

Top minus bottom.

∫ from 2 to 4 of f(x) dx = F(4) − F(2)

= (4²2 − 2(4)) − (2²2 − 2(2))

= 0 − (−2)

That is 2.

= 2

Careful!

Not one integral from 0 to 4!

The piece below the axis counts as negative, so the pieces cancel.

Careful: not area = ∫ from 0 to 4 of f(x) dx = 0 ✗

5 Add the areas

−2 is below the axis.

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

below: area = 2

Add the areas of the pieces.

area = 2 + 2

So the area is 4 square units.

= 4 square units

6 Check

Let's check the pieces: the whole integral from 0 to 4 is 0.

F(4) − F(0) = 0

The signed pieces add up to the same: −2 plus 2.

Right!

−2 + 2 = 0

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