There are two ways of thinking about
integration, and they both have their uses. The one we shall
concentrate on here, is integration as the
inverse of differentiation,
also call indefinite integration,
Indefinite integral
where is the inverse derivative (also called “primitive”) of .
Example 5.1:
Integrate .
Solution:
, so
This type of integration is called an indefinite integral. We always get a constant of integration (in this case “ ”) for an indefinite integral.
Note: The result of
is another function of
.
Definite integral
A definite integral is related to
the area under a curve (see Fig.
??)
Plot the curve , as in Fig. 5.1 . The shaded area under curve between and equal a number . We can calculate this as
(This is called a definite integral.) This is defined as the sum from to of the area of all the small strips under the curve, in the limit that they become vanishingly thin.
The two definitions are related by the
Fundamental theorem of calculus
Example 5.2:
Find area under curve between and .
Solution:
The area is given by the integral
Note: There is no constant of integration in a definite integral.
Note: The result is a number not a function.
Final Remark: Some integrals can never be done in terms of known functions.
Example 5.3:
, . For these a numerical method will give results for a definite integral, e.g., a computer version of summing the area of the strips under a curve.
We can use the “inverse derivative” to look up standard derivatives from right to left, to get a table of integrals. Many of the integrals in the formula book were obtained this way, Some examples:
Let us look at an interesting physics example of integration.
Example 5.4:
A piston is moved in a cylinder containing an ideal gas (see Fig. 5.2 ). Calculate the work done as the enclosed volume changes.
Solution:
The work done is force times distance. The maginitude of the force is pressure times ares, . If we move the piston by a distance , the work done is thus . Thus
For an ideal gas , and we find that
Note that the argument of the logarithm is dimensionless. This is true of any mathematical functions we write.