Step by step
The question
Question: find the area between y = x² − 1 and the x-axis from x = 0 to x = 3.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw the region
Draw the curve first.
2 Where it crosses
Where does it cross the x-axis?
Where y is 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).
So it factors like this.
So x = −1 or x = 1.
From 0 to 1 it is below the axis: shaded red.
From 1 to 3 the curve is above the axis: shaded green.
An integral counts area below the x-axis as negative.
But an area is never negative.
So we work out each piece on its own.
3 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
4 Each piece
The first piece: from 0 to 1.
Top minus bottom.
Put in 1, then take away the same with 0.
First the powers and the times.
Then each bracket.
1 is 3/3, so the same bottom.
The first is −2/3.
The second is 0.
That is −2/3.
Now the second piece: from 1 to 3.
Top minus bottom.
Put in 3, then take away the same with 1.
First the powers and the times.
Then each bracket.
The first is 6.
1 is 3/3, so the same bottom.
The second is −2/3.
Taking away a minus is the same as adding.
18 plus 2 is 20, over 3.
That is 20/3.
Careful!
Not one integral from 0 to 3!
The piece below the axis counts as negative, so the pieces cancel.
5 Add the areas
−2/3 is below the axis.
Its area is the size, without the minus: 2/3.
Add the areas of the pieces.
So the area is 22/3 square units.
6 Check
Let's check the pieces: the whole integral from 0 to 3 is 6.
The signed pieces add up to the same: −2/3 plus 20/3.
Right!
7 Pick the choice
Our answer is area = 22/3 square units.
That is choice C.
8 Why not the others
A is 6.
Wrong: ∫ from 0 to 3: the part under the axis takes area away.
That is the Careful mistake from before.
B is 20/3.
Wrong: only the part above the axis.
D is 8.
Wrong: put 3 into x² − 1.
Let's look back.
First: draw the region.
Then: where it crosses.
Then: the antiderivative.
Then: each piece.
Then: add the areas.
Then: check.
Then: pick the choice.
Last: why not the others.