What is the area of the region between y = 1 and y = x^2? Step by Step

Step by step

The question

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

y = 1

There are four choices.

First we solve it, the normal way.

Then we pick.

y = x²

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.

1 = x²

We want 0 on one side.

So we take 1 away from both sides, and write the two sides the other way round.

x² − 1 = 0

Factor it.

1 is 1 times 1.

So x² − 1 is x² minus 1²: one square take away another square.

The rule: a² − b² = (a − b)(a + b).

So it is (x − 1)(x + 1).

1 = 1 × 1

= 1²

a² − b² = (a − b)(a + b)

So it factors like this.

(x + 1)(x − 1) = 0

So x = −1 or x = 1.

x = −1 or x = 1

3 The top curve

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

The first gives 1, the second 0.

So y = 1 is on top.

x = 0: 1 > 0

The area between them is shaded.

4 Write the integral

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

At each x, top minus bottom is the height of the gap between them.

area = ∫ from −1 to 1 of ((1) − x²) dx

Careful!

Top minus bottom!

The other way gives a minus answer, −4/3.

Careful: not bottom − top ✗

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

Then like terms.

= ∫ from −1 to 1 of (−x² + 1) dx

5 The antiderivative

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

F(x) = −x³3 + x

6 Top minus bottom

Top minus bottom: F(1) − F(−1).

area = F(1) − F(−1)

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

= (−1³3 + 1) − (−(−1)³3 + (−1))

First the powers and the times.

= (−13 + 1) − (13 − 1)

Then each bracket.

1 is 3/3, so the same bottom.

The first is 2/3.

1 is 3/3, so the same bottom.

The second is −2/3.

= 23 − (−23)

Taking away a minus is the same as adding.

− (−) = +

2 plus 2 is 4, over 3.

So the area is 4/3 square units.

= 43 square units

7 Check

Let's check F: its derivative is −x² + 1: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + 1

8 Pick the choice

Our answer is area = 4/3 square units.

That is choice B.

B) 43

9 Why not the others

A is 2/3.

Wrong: used x from 0 to 1 only: the curves meet at −1 and 1.

A) 23 ✗ used x from 0 to 1 only: the curves meet at −1 and 1

C is 2.

Wrong: only the area under y = 1.

C) 2 ✗ only the area under y = 1

D is 8/3.

Wrong: added the area under y = x²: take it away.

D) 83 ✗ added the area under y = x^2: take it away

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.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer