5.5 Uses of the integral calculus 147
(four times the length of that quarter in the first quadrant, see Example 5.11). The
integral is one of the standard integrals listed in Table 6.3:
The principal values of the inverse sine of 1 and of 0 are π 22 and 0 respectively.
Therefores 1 = 12 πa.
0 Exercise 49
5.5 Uses of the integral calculus
The discussion of Section 5.4 was concerned with the use of the integral calculus
for the determination of geometric properties; in particular, the area of a plane figure
and the length of a curve in a plane. The generalization to curves in three dimensions,
to nonplanar surfaces, and to volumes is discussed in Chapters 9 and 10.
We saw in Section 5.3, equation (5.13), that the definite integral provides the
average value of a function. When the function represents, for example, the distri-
bution of mass in a physical body, the integral calculus can be used to determine
static properties of the body such as the total mass, the centre of mass, and the
moment of inertia in terms of definite integrals involving the mass density.
Similarly for a distribution of charge or of any physical property of a system that
is distributed in space. This use of the integral calculus for the determination of
static properties is introduced in Section 5.6 for properties distributed along a line
(the one-dimensional case); the three-dimensional case is discussed in Chapter 10.
The same methods are used in Chapter 21 for the analysis of probability distributions
in statistics.
The concept of the integral was introduced in Section 5.1 by considering the
motion of a body along a line. By Newton’s first law of motion, a body moving with
a given velocity at any time continues to move along a straight line with the same
velocity if no external forces act on the body; v 1 = 1 constantwhen F 1 = 10. On the
other hand, by Newton’s second law, the acceleration, or rate of change of velocity
a 1 = 1 dv 2 dt, experienced by a body of mass min the presence of an external force Fis
given by F 1 = 1 ma. The use of the integral calculus for the description of the dynamics
of physical systems is introduced in Section 5.7 for motion along a straight line;
the more general case of motion in two and three dimensions is discussed in
Chapter 16. The application to pressure–volume work in thermodynamics is
discussed in Section 5.8.
The widest use of the integral calculus is for the solution (integration) of differential
equations; in particular, the equations of chemical kinetics and of other rate processes,
the equations of motion in classical mechanics (derived from Newton’s second
law for example), and the equations of motion in quantum mechanics (such as the
Schrödinger equation). This use of the integral calculus is discussed in Chapters 11
to 14.
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