154
Appendix A: Mathematics
Fig. A.1 Graphical
representation of the definite
integral as the area A under
the graph of f (X )
−5
0
5
0
5
10
15
20
A
X-coordinate
Function f (X) = X
2
+ 1
A.1 Example: Indefinite and definite integral
Calculate the indefinite and definite integral of f (X ) = X
2
+ 1. The definite integral
is to be calculated in the interval [1, 2]. Furthermore, give a graphical interpretation
of the definite integral.
A.1 Solution
Indefinite Integral:
F(X ) =
(X
2
+ 1)dX =
X
3
3
+ X + c .
(A.7)
Definite Integral:
2
1
(X
2
+ 1)dX =
X
3
3
+ X
2
1
=
10
3
.
(A.8)
The graphical interpretation of the definite integral is shown in Fig. A.1.