Area between y = x^2 - 6x + 5 and y = x

Step by step

The question

Question: find the area between y = x² − 6x + 5 and y = x − 5.

y = x² − 6x + 5

y = x − 5

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 − 6x + 5 = x − 5

Move everything to one side.

x² − 7x + 10 = 0

Factor it.

We need two numbers that multiply to 10 and add to −7: −2 and −5.

(x − 2)(x − 5) = 0

So x = 2 or x = 5.

x = 2 or x = 5

3 The top curve

Test a number between 2 and 5: x = 4.

The first gives −3, the second −1.

So y = x − 5 is on top.

x = 4: −3 < −1

The area between them is shaded.

4 Write the integral

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

area = ∫ from 2 to 5 of ((x − 5) − (x² − 6x + 5)) 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 2 to 5 of (−x² + 7x − 10) dx

5 The antiderivative

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

F(x) = −x³3 + 7 × x²2 − 10x

Now we have F(x) = −x³ over 3 + 7x² over 2 − 10x.

= −x³3 + 7x²2 − 10x

6 Top minus bottom

area = F(5) − F(2)

Put in 5, then take away the same with 2.

= (−5³3 + 7(5)²2 − 10(5)) − (−2³3 + 7(2)²2 − 10(2))

= −256 − (−263)

So the area is 9/2 square units.

= 92 square units

7 Check

Let's check F: its derivative is −x² + 7x − 10: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + 7x − 10

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