Step by step
The question
Question: find the area between y = 4x − 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.
Every term has −x, so we take it out.
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) = 2 x² − 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 0 and 4 on the bar.
Put in 4 first, then take away the same with 0.
First the powers and the times.
Then each bracket: the first is 32/3, the second is 0.
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 4x − x², the function we started with.
Right!
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: top minus bottom.
Last: check.