Area between y = x^2 and y = x + 6 Step by Step

Step by step

The question

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

y = x²

y = x + 6

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 = x + 6

Move everything to one side.

x² − x − 6 = 0

Factor it.

We need two numbers that multiply to −6 and add to −1: 2 and −3.

(x + 2)(x − 3) = 0

So x = −2 or x = 3.

x = −2 or x = 3

3 The top curve

Test a number between −2 and 3: x = 1.

The first gives 1, the second 7.

So y = x + 6 is on top.

x = 1: 1 < 7

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 3 of ((x + 6) − x²) dx

Careful!

Top minus bottom!

The other way gives a minus answer, −125/6.

Careful: not bottom − top ✗

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

Then like terms.

= ∫ from −2 to 3 of (−x² + x + 6) dx

5 The antiderivative

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

F(x) = −x³3 + x²2 + 6x

6 Top minus bottom

area = F(3) − F(−2)

Put in 3, then take away the same with −2.

= (−3³3 + 3²2 + 6(3)) − (−(−2)³3 + (−2)²2 + 6(−2))

= 272 − (−223)

So the area is 125/6 square units.

= 1256 square units

7 Check

Let's check F: its derivative is −x² + x + 6: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + x + 6

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