Step by step
The question
Question: find the integral from 2 to 4 of x − 1.
∫ from 2 to 4 of (x − 1) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = x² over 2 − x.
F(x) = x22 − x
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 2 and 4 on the bar.
= (x22 − x) from 2 to 4
Put in 4 first, then take away the same with 2.
= (422 − 4) − (222 − 2)
Work out each bracket.
= 4 − 0
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(2) − F(4) ✗
So the integral is 4.
= 4
3 Check
Let's check with geometry.
Under a straight line the shape is a trapezoid: heights 1 and 3, width 2.
Its area is 4.
The same!
Right!
trapezoid: 1 + 32 × 2 = 4
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
Answer
= 4