Step by step
The question
Question: find the integral from 0 to 2 of x² − 4.
∫ from 0 to 2 of (x2 − 4) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x³ over 3 − 4x.
F(x) = x33 − 4x
No + C here: in F(b) minus F(a), the two Cs cancel.
no + C here: it cancels
2 Top minus bottom
The rule: put in the top number, then take away the bottom number.
∫ from a to b of f(x) dx = F(b) − F(a)
Write F(x) with 0 and 2 on the bar.
= (x33 − 4x) from 0 to 2
Put in 2 first, then take away the same with 0.
= (233 − 4(2)) − (033 − 4(0))
Work out each bracket.
= −163 − 0
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(0) − F(2) ✗
So the integral is −16/3.
= −163
3 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: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= −163