Area between y = x^2 - 1 and y = x + 1 Step by Step

Step by step

The question

Question: find the area between y = x² − 1 and y = x + 1.

y = x² − 1

y = x + 1

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 − 1 = x + 1

Move everything to one side.

x² − x − 2 = 0

Factor it.

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

(x + 1)(x − 2) = 0

So x = −1 or x = 2.

x = −1 or x = 2

3 The top curve

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

The first gives 0, the second 2.

So y = x + 1 is on top.

x = 1: 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 −1 to 2 of ((x + 1) − (x² − 1)) 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 −1 to 2 of (−x² + x + 2) dx

5 The antiderivative

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

F(x) = −x³3 + x²2 + 2x

6 Top minus bottom

area = F(2) − F(−1)

Put in 2, then take away the same with −1.

= (−2³3 + 2²2 + 2(2)) − (−(−1)³3 + (−1)²2 + 2(−1))

= 103 − (−76)

So the area is 9/2 square units.

= 92 square units

7 Check

Let's check F: its derivative is −x² + x + 2: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + x + 2

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