Step by step
The question
Question: find the area between y = 1 − x² and the x-axis.
1 Draw the region
Draw the curve first.
Where does it cross the x-axis?
Where y is 0.
Factor it.
x² − 1 is a difference of two squares: x² minus 1 squared.
So it is (x − 1)(x + 1).
So it crosses at x = −1 and x = 1.
The area we want is shaded: under the curve, above the x-axis, from −1 to 1.
This area is the integral from −1 to 1.
2 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x − x³ over 3.
No + C here: in F(b) minus F(a), the two Cs cancel.
3 Top minus bottom
The rule: put in the top number, then take away the bottom number.
Write F(x) with −1 and 1 on the bar.
Put in 1 first, then take away the same with −1.
Work out each bracket.
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
So the area is 4/3 square units.
4 Check
Let's check F: its derivative is 1 − x², the function we started with.
Right!
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: top minus bottom.
Last: check.