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

Step by step

The question

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

y = 6 − x²

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.

6 − x² = x

Move everything to one side.

x² + x − 6 = 0

Factor it.

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

(x + 3)(x − 2) = 0

So x = −3 or x = 2.

x = −3 or x = 2

3 The top curve

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

The first gives 6, the second 0.

So y = 6 − x² is on top.

x = 0: 6 > 0

The area between them is shaded.

4 Write the integral

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

area = ∫ from −3 to 2 of ((6 − x²) − x) 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 −3 to 2 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(2) − F(−3)

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

= (−2³3 − 2²2 + 6(2)) − (−(−3)³3 − (−3)²2 + 6(−3))

= 223 − (−272)

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