Step by step
The question
Question: find the integral from 0 to 2 of 3x².
∫ from 0 to 2 of 3x2 dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
F(x) = 3 × x33
Now we have F(x) = x³.
= x3
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.
= (x3) from 0 to 2
Put in 2 first, then take away the same with 0.
= (23) − (03)
Work out each bracket.
= 8 − 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 8.
= 8
3 Check
Let's check F: its derivative is 3 x², the function we started with.
Right!
F′(x) = 3x2
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 8