Step by step
The question
Question: find the area between y = −x² and the x-axis from x = 0 to x = 2.
y = −x2
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
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) − (−033)
= −83 − 0
That is −8/3.
= −83
Careful!
An area is never negative.
The minus only says: below the axis.
Careful: not area = −83 ✗
4 Add the areas
−8/3 is below the axis.
Its area is the size, without the minus: 8/3.
below: area = 83
So the area is 8/3 square units.
area = 83 square units
5 Check
Let's check F: its derivative is −x², the function we started with.
Right!
F′(x) = −x2
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: each piece.
Then: add the areas.
Last: check.
Answer
area = 83 square units