Worked examples
The question
Question: find the integral from 1 to 3 of 2x + 1.
∫ from 1 to 3 of (2x + 1) dx =
1 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
F(x) = 2 × x22 + x
Now we have F(x) = x² + x.
= x2 + 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 1 and 3 on the bar.
= (x2 + x) from 1 to 3
Put in 3 first, then take away the same with 1.
= (32 + 3) − (12 + 1)
Work out each bracket.
= 12 − 2
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 10.
= 10
3 Check
Let's check with geometry.
Under a straight line the shape is a trapezoid: heights 3 and 7, width 2.
Its area is 10.
The same!
Right!
trapezoid: 3 + 72 × 2 = 10
Let's look back.
First: the antiderivative.
Then: top minus bottom.
Last: check.
▶ Watch it solved
The question
Question: find the integral from 1 to 2 of 3x².
∫ from 1 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 1 and 2 on the bar.
= (x3) from 1 to 2
Put in 2 first, then take away the same with 1.
= (23) − (13)
Work out each bracket.
= 8 − 1
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(1) − F(2) ✗
So the integral is 7.
= 7
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.
▶ Watch it solved
The question
Question: find the area between y = 4 − x² and the x-axis.
y = 4 − x2
1 Draw the region
Draw the curve first.
Where does it cross the x-axis?
Where y is 0.
4 − x² = 0
Factor it.
x² − 4 is a difference of two squares: x² minus 2 squared.
So it is (x − 2)(x + 2).
−(x + 2)(x − 2) = 0
So it crosses at x = −2 and x = 2.
x = −2 or x = 2
The area we want is shaded: under the curve, above the x-axis, from −2 to 2.
This area is the integral from −2 to 2.
area = ∫ from −2 to 2 of (4 − x2) dx
2 The antiderivative
First the antiderivative: add 1 to each power, divide by the new power.
Now we have F(x) = 4x − x³ over 3.
F(x) = 4x − x33
No + C here: in F(b) minus F(a), the two Cs cancel.
no + C here: it cancels
3 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 2 on the bar.
= (4x − x33) from −2 to 2
Put in 2 first, then take away the same with −2.
= (4(2) − 233) − (4(−2) − (−2)33)
Work out each bracket.
= 163 − (−163)
Careful!
Top first, then take away the bottom.
The other way gives the wrong sign.
Careful: not F(−2) − F(2) ✗
So the area is 32/3 square units.
area = 323 square units
4 Check
Let's check F: its derivative is 4 − x², the function we started with.
Right!
F′(x) = 4 − x2
Let's look back.
First: draw the region.
Then: the antiderivative.
Then: top minus bottom.
Last: check.
Answer area = 323 square units
▶ Watch it solved
