What is the area of the region between y = x^2 + 1 and y = 2x + 4? Step by Step

Step by step

The question

Question: find the area between y = x² + 1 and y = 2x + 4.

y = x² + 1

There are four choices.

First we solve it, the normal way.

Then we pick.

y = 2x + 4

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 = 2x + 4

We want 0 on one side.

So we take 2x + 4 away from both sides.

x² − 2x − 3 = 0

Factor it.

To factor x² − 2x − 3, the rule: we need two numbers that multiply to −3 and add to −2.

They are 1 and −3: they multiply to −3 and add to −2.

? × ? = −3

? + ? = −2

1 × (−3) = −3, 1 + (−3) = −2 ✓

So it factors like this.

(x + 1)(x − 3) = 0

So x = −1 or x = 3.

x = −1 or x = 3

3 The top curve

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

The first gives 2, the second 6.

So y = 2x + 4 is on top.

x = 1: 2 < 6

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 3 of ((2x + 4) − (x² + 1)) dx

Careful!

Top minus bottom!

The other way gives a minus answer, −32/3.

Careful: not bottom − top ✗

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

Then like terms.

= ∫ from −1 to 3 of (−x² + 2x + 3) dx

5 The antiderivative

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

F(x) = −x³3 + 2 × x²2 + 3x

Now we have F(x) = −x³ over 3 + x² + 3x.

= −x³3 + x² + 3x

6 Top minus bottom

Top minus bottom: F(3) − F(−1).

area = F(3) − F(−1)

Put in 3, then take away the same with −1.

= (−3³3 + 3² + 3(3)) − (−(−1)³3 + (−1)² + 3(−1))

First the powers and the times.

= (−9 + 9 + 9) − (13 + 1 − 3)

Then each bracket.

The first is 9.

1 is 3/3, and 3 is 9/3, so the same bottom.

The second is −5/3.

= 9 − (−53)

Taking away a minus is the same as adding.

− (−) = +

27 plus 5 is 32, over 3.

So the area is 32/3 square units.

= 323 square units

7 Check

Let's check F: its derivative is −x² + 2x + 3: top minus bottom.

And the area is positive.

Right!

F′(x) = −x² + 2x + 3

8 Pick the choice

Our answer is area = 32/3 square units.

That is choice D.

D) 323

9 Why not the others

A is 9.

Wrong: used x from 0 to 3: the curves meet at −1 and 3.

A) 9 ✗ used x from 0 to 3: the curves meet at −1 and 3

B is 24.

Wrong: only the area under the line.

B) 24 ✗ only the area under the line

C is −32/3.

Wrong: took the top curve away from the bottom one.

That is the Careful mistake from before.

C) −323 ✗ took the top curve away from the bottom one

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