Step by step
The question
Question: find the integral from 0 to 3 of 2x.
∫ from 0 to 3 of 2x dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
F(x) = 2 × x22
Now we have F(x) = x².
= x2
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 3 on the bar.
= (x2) from 0 to 3
Put in 3 first, then take away the same with 0.
= (32) − (02)
Work out each bracket.
= 9 − 0
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(0) − F(3) ✗
So the integral is 9.
= 9
3 Check
Let's check with geometry.
Under a straight line the shape is a trapezoid: heights 0 and 6, width 3.
Its area is 9.
The same!
Right!
trapezoid: 0 + 62 × 3 = 9
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 9