Step by step
The question
Question: find the area between y = x² and y = x + 2.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
We want 0 on one side.
So we take x + 2 away from both sides.
Factor it.
To factor x² − x − 2, the rule: we need two numbers that multiply to −2 and add to −1.
They are 1 and −2: they multiply to −2 and add to −1.
So it factors like this.
So x = −1 or x = 2.
3 The top curve
Test a number between −1 and 2: x = 1.
The first gives 1, the second 3.
So y = x + 2 is on top.
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.
Careful!
Top minus bottom!
The other way gives a minus answer, −9/2.
A minus in front of a bracket changes the sign of every part inside.
Then like terms.
5 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
6 Top minus bottom
Top minus bottom: F(2) − F(−1).
Put in 2, then take away the same with −1.
First the powers and the times.
We count up in 3s and 2s.
In 3s: 3, 6.
In 2s: 2, 4, 6.
6 is the first number in both lists.
So 6 works for both.
Then each bracket.
2 is 6/3, and 4 is 12/3, so the same bottom.
The first is 10/3.
1/3 is 2/6, and 1/2 is 3/6, and 2 is 12/6, so the same bottom.
The second is −7/6.
We count up in 3s and 6s.
In 3s: 3, 6.
In 6s: 6.
6 is the first number in both lists.
So 6 works for both.
10/3 is 20/6, so the same bottom.
Taking away a minus is the same as adding.
20 plus 7 is 27, over 6.
To make it simpler: What goes into 27 and 6?
We list the numbers that go into each.
Into 27: 1, 3, 9, 27.
Into 6: 1, 2, 3, 6.
3 is the biggest one in both lists.
So the area is 9/2 square units.
7 Check
Let's check F: its derivative is −x² + x + 2: top minus bottom.
And the area is positive.
Right!
8 Pick the choice
Our answer is area = 9/2 square units.
That is choice D.
9 Why not the others
A is 7.5.
Wrong: only the area under the line.
B is 3.
Wrong: only the area under the parabola.
C is 10.5.
Wrong: added the two areas: take one 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.