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

Step by step

The question

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

y = x² − 2x

y = x

1 Draw the region

Draw both curves.

They cross twice.

The area we want is between the crossings.

2 Where they meet

They meet where the two y's are equal.

x² − 2x = x

Move everything to one side.

x² − 3x = 0

Factor it.

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

x(x − 3) = 0

So x = 0 or x = 3.

x = 0 or x = 3

3 The top curve

Test a number between 0 and 3: x = 2.

The first gives 0, the second 2.

So y = x is on top.

x = 2: 0 < 2

The area between them is shaded.

4 Write the integral

Area between = the integral of the top one minus the bottom one.

area = ∫ from 0 to 3 of ((x) − (x² − 2x)) dx

Careful!

Top minus bottom!

The other way gives a minus answer, −9/2.

Careful: not bottom − top ✗

A minus in front of a bracket changes the sign of every part inside.

Then like terms.

= ∫ from 0 to 3 of (−x² + 3x) dx

5 The antiderivative

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

F(x) = −x³3 + 3 × x²2

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

= −x³3 + 3x²2

6 Top minus bottom

area = F(3) − F(0)

Put in 3, then take away the same with 0.

= (−3³3 + 3(3)²2) − (−0³3 + 3(0)²2)

= 92 − 0

So the area is 9/2 square units.

= 92 square units

7 Check

Let's check F: its derivative is −x² + 3x: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + 3x

Let's look back.

First: draw the region.

Then: where they meet.

Then: the top curve.

Then: write the integral.

Then: the antiderivative.

Then: top minus bottom.

Last: check.

Answer