This document provides an introduction to basic definite integration. It defines definite integration as calculating the area under a curve between two limits using antiderivatives. It demonstrates how to calculate definite integrals of simple functions and interpret the results as areas. It also discusses how the sign of the integral depends on whether the function lies above or below the x-axis. Quizzes are included to assess understanding of these concepts.

1. Basic Mathematics
De

2. nite Integration
R Horan & M Lavelle
The aim of this package is to provide a short self
assessment programme for students who want to
be able to calculate basic de

3. nite integrals.
Copyright 2004 rhoran@plymouth.ac.uk , mlavelle@plymouth.ac.uk
c
Last Revision Date: September 9, 2005 Version 1.0

4. Table of Contents
1. Introduction
2. De

5. nite Integration
3. The Area Under a Curve
4. Final Quiz
Solutions to Exercises
Solutions to Quizzes
The full range of these packages and some instructions,
should they be required, can be obtained from our web
page Mathematics Support Materials.

6. Section 1: Introduction 3
1. Introduction
It is possible to determine a function F (x) from its derivative f (x) by
calculating the anti-derivative or integral of f (x), i.e.,

dF
if = f (x) ; then F (x) = f (x)dx + C
dx
where C is an integration constant (see the package on inde

7. nite
integration). In this package we will see how to use integration to
calculate the area under a curve.
As a revision exercise, try this quiz on inde

8. nite integration.
‚
Quiz Select the inde

9. nite integral of (3x2 1 x)dx with respect to x
2
1 3 3
(a) 6x + C ; (b) x x2 + C ;
2 2
1 2 1
(c) x + x + C ;
2
(d) x3 x2 + C :
4 4
Hint: If n T= 1 ; the integral of x is x
n n+1 =(n + 1).

10. Section 2: De

11. nite Integration 4
2. De

12. nite Integration
We de

13. ne the de

14. nite integral of the function f (x) with respect to
x from a to b to be
 b b
f (x)dx = F (x) = F (b) F (a) ;
a a
where F (x) is the anti-derivative of f (x). We call a and b the lower
and upper limits of integration respectively. The function being inte-
grated, f (x), is called the integrand. Note the minus sign!
Note integration constants are not written in de

15. nite integrals since
they always cancel in them:
 b b
f (x)dx = F (x)
a a
= (F (b) + C ) (F (a) + C )
= F (b) + C F (a) C
= F (b) F (a) :

16. Section 2: De

17. nite Integration 5
‚2
Example 1 Calculate the de

18. nite integral 1
x3 dx.
‚ a
From the rule axn dx = xn+1 we have
n+1

2
1 3+1 2
3
x dx = x
1 3+1 1
2
1 1 1
= x4 = ¢ 24 ¢ 11
4 1 4 4
1
=
4
¢ 16 1 = 4 1 = 15 :
4 4 4
Exercise 1. Calculate the following de

19. nite integrals: (click on the
green letters for the solutions)
‚3 ‚2
(a) 0
xdx ; (b) 1 xdx ;
‚2 ‚2
(c) 1
(x2 x)dx ; (d) 1 (x
2
x)dx :

20. Section 3: The Area Under a Curve 6
3. The Area Under a Curve
The de

21. nite integral of a function f (x) which lies above the x axis
can be interpreted as the area under the curve of f (x).
Thus the area shaded blue below
y
y = f (x)
A
x
0 a b
is given by the de

22. nite integral
 b b
f (x)dx = F (x)
= F (b) F (a) :
a a
This is demonstrated on the next page.

23. Section 3: The Area Under a Curve 7
Consider the area, A, under the curve, y = f (x). If we increase
the value of x by x, then the increase in area, A, is approximately
y
A = y x A A = y :
x
Here we approximate the y
area of the thin strip by
a rectangle of width x
and height y . In the limit A A
as the strips become thin, x x+x
x
x 3 0, this means:
0
dA A
= lim = y:
dx x 30 x
The function (height of the curve) is the derivative of the area and
the area below the curve is an anti-derivative or integral of
the function.
N.B. so far we have assumed that y = f (x) lies above the x axis.

24. Section 3: The Area Under a Curve 8
‚3
Example 2 Consider the integral 0
xdx. The integrand y = x (a
straight line) is sketched below. The area underneath the line is the
blue shaded triangle. The area of any triangle is half its base times
the height. For the blue shaded triangle, this is
1 9
A= ¢3¢3= :
2 2
y
y =x
3
A
x
0 3
As expected, the integral yields the same result:
 3 3
= 3 0 = 9 0= 9:
x2 2 2
xdx =
0 2 0 2 2 2 2

25. Section 3: The Area Under a Curve 9
Here is a quiz on this relation between de

26. nite integrals and the area
under a curve.
Quiz Select the value of the de

27. nite integral
 3
2dx ;
1
which is sketched in the following diagram:
y
y =2
2
A
x
0 1 3
(a) 6 ; (b) 2 ; (c) 4 ; (d) 8 :
Hint: 2 may be written as 2x , since x = 1.
0 0

28. Section 3: The Area Under a Curve 10
Example 3 Consider the two lines: y = 3 and y = 3.
Let us integrate these functions in y 6
turn from x = 0 to x = 2. y = +3
3
a) For y = 3:
-
 2
2
(+3)dx = 3x = 3¢2 3¢0 = 6 : 0 j j
0
0 1 2 x
and 6 is indeed the area of the rect- 3
angle of height 3 and length 2. y = 3
b) However, for y = 3:
 2
2
( 3)dx = 3x
= 3¢2 ( 3¢0) = 6 :
0 0
Although both rectangles have the same area, the sign of this result
is negative because the curve, y = 3, lies below the x axis. This
indicates the sign convention:
If a function lies below the x axis, its integral is negative.
If a function lies above the x axis, its integral is positive.

29. Section 3: The Area Under a Curve 11
Exercise 2.
y 6
y1 (x)
A B 0 C D
-
x
y2 (x)
From the diagram above, what can you say about the signs of the
following de

30. nite integrals? (Click on the green letters for the solu-
tions)
‚B ‚D
(a) A y1 (x)dx ; (b) B y1 (x)dx ;
‚0 ‚D
(c) A y2 (x)dx ; (d) C y2 (x)dx :

31. Section 3: The Area Under a Curve 12
‚ ‚ a
Example 4 To calculate 42 6x2 dx, use axn dx = xn+1 . Thus
n+1
 2 2
6
6x2 dx = x2+1
4 2+1 4
6 2 2
= x3 = 2x3
3 4
4
= 2 ¢ ( 2)3 2 ¢ ( 4)3 = 16 + 128 = 112 :
Note that even though the integration range is for negative x (from
4 to 2), the integrand, f (x) = 6x2 , is a positive function. The
de

32. nite integral of a positive function is positive. (Similarly it is
negative for a negative function.)
Quiz Select the de

33. nite integral of y = 5x4 with respect to x if the
lower limit of the integral is x = 2 and the upper limit is x = 1
(a) 31 ; (b) 31 ; (c) 29 ; (d) 27 :

34. Section 3: The Area Under a Curve 13
Exercise 3. Use the integrals listed below to calculate the following
de