The Chemistry Maths Book, Second Edition

136 Chapter 5Integration

Figure 5.6 shows that the integrand, sin 1 x, has positive values when 01 < 1 x 1 < 1 π, and

negative values whenπ 1 < 1 x 1 < 12 π. The total ‘area under the curve’ is the sum of the two

areas labelled A

1

and A

2

in the figure, and the integral can be written as

where

This example shows that areas corresponding to negative values of the integrand

make negative contributions to the total. In this case the positive and negative

contributions cancel, and the average value of sin 1 xin the interval is zero.

0 Exercise 31

Integration of discontinuous functions

The function

is discontinuous at x 1 = 12 , but Figure 5.7 shows

that the function can be integrated across the

discontinuity if the range of integration is split at

the point of discontinuity. Thus,

This can always be done when the integrand has only a finite number of finite

discontinuities within the range of integration. A similar technique is used when the

function is continuous but has a discontinuous derivative.

=++=+=ZZ

1

2

2

3

221369 xdx ()x dx

ZZZ

1

3

1

2

2

3

fxdx fxdx fxdx() =+() ()

fx

x

x

x

x

()=

• 
  











<

≥

2

21

2

2

if

if

A x dx A x dx

1

0

2

2

==+ZZ 22 = =−

π

π

π

sin and sin

Z

0

2

12

π

sinxdx A A=+

• 1

− 1

O π 2 π

A

1

A

2

x

si nx

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Figure 5.6

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Figure 5.7