Step by step
The question
Question: find the area between y = −x² + 4x 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.
We need two numbers that multiply to 0 and add to −4: 0 and −4.
So it crosses at x = 0 and x = 4.
The area we want is shaded: under the curve, above the x-axis, from 0 to 4.
This area is the integral from 0 to 4.
2 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = −x³ over 3 + 2 x².
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 0 and 4 on the bar.
Put in 4 first, then take away the same with 0.
Work out each bracket.
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
So the area is 32/3 square units.
4 Check
Let's check F: its derivative is − x² + 4x, the function we started with.
Right!
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: top minus bottom.
Last: check.