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

Step by step

The question

Question: find the area between y = 4 − x² and y = x + 2.

y = 4 − x²

y = x + 2

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.

4 − x² = x + 2

Move everything to one side.

x² + x − 2 = 0

Factor it.

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

(x + 2)(x − 1) = 0

So x = −2 or x = 1.

x = −2 or x = 1

3 The top curve

Test a number between −2 and 1: x = 0.

The first gives 4, the second 2.

So y = 4 − x² is on top.

x = 0: 4 > 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 −2 to 1 of ((4 − x²) − (x + 2)) 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 1 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(1) − F(−2)

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

= (−1³3 − 1²2 + 2(1)) − (−(−2)³3 − (−2)²2 + 2(−2))

= 76 − (−103)

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