Area between y = x^2 - 4x and the x-axis Step by Step

Step by step

The question

Question: find the area between y = x² − 4x and the x-axis.

y = x² − 4x

1 Draw the region

Draw the curve first.

2 Where it crosses

Where does it cross the x-axis?

Where y is 0.

x² − 4x = 0

Factor it.

We need two numbers that multiply to 0 and add to −4: 0 and −4.

x(x − 4) = 0

So x = 0 or x = 4.

x = 0 or x = 4

From 0 to 4 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 − 4 × x²2

Now we have F(x) = x³ over 3 − 2 x².

= x³3 − 2x²

4 Each piece

The integral: from 0 to 4.

Top minus bottom.

∫ from 0 to 4 of f(x) dx = F(4) − F(0)

= (4³3 − 2(4)²) − (0³3 − 2(0)²)

= −323 − 0

That is −32/3.

= −323

Careful!

An area is never negative.

The minus only says: below the axis.

Careful: not area = −323 ✗

5 Add the areas

−32/3 is below the axis.

Its area is the size, without the minus: 32/3.

below: area = 323

So the area is 32/3 square units.

area = 323 square units

6 Check

Let's check F: its derivative is x² − 4x, the function we started with.

Right!

F′(x) = x² − 4x

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