Step by step
The question
Question: find the integral from 1 to 3 of x² − 4.
∫ from 1 to 3 of (x2 − 4) dx
An integral from a to b gives one number.
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 1 and 3 on the bar.
= (x33 − 4x) from 1 to 3
Put in 3 first, then take away the same with 1.
= (333 − 4(3)) − (133 − 4(1))
First the powers and the times.
= (9 − 12) − (13 − 4)
Then each bracket: the first is −3, the second is −11/3.
= −3 − (−113)
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(1) − F(3) ✗
So the integral is 2/3.
= 23
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
= 23