Step by step
The question
Question: find the area between y = x² − 4 and the x-axis from x = 0 to x = 2.
y = x2 − 4
1 Draw the region
Draw the curve first.
From 0 to 2 it is below the axis: shaded red.
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.
∫ below the x-axis → a minus number
2 The antiderivative
The antiderivative: add 1 to each power, divide by the new power.
F(x) = x33 − 4x
3 Each piece
The integral: from 0 to 2.
Top minus bottom.
∫ from 0 to 2 of f(x) dx = F(2) − F(0)
= (233 − 4(2)) − (033 − 4(0))
= −163 − 0
That is −16/3.
= −163
Careful!
An area is never negative.
The minus only says: below the axis.
Careful: not area = −163 ✗
4 Add the areas
−16/3 is below the axis.
Its area is the size, without the minus: 16/3.
below: area = 163
So the area is 16/3 square units.
area = 163 square units
5 Check
Let's check F: its derivative is x² − 4, the function we started with.
Right!
F′(x) = x2 − 4
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: each piece.
Then: add the areas.
Last: check.
Answer
area = 163 square units