Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

abx 1 ξ 1 x 2 ξ 2 x 3 ξ 3 x 4 ξ 4 x 5

f(x)

x

Figure 2.8 The evaluation of a definite integral by subdividing the interval

a≤x≤binto subintervals.

Evaluate from first principles the integralI=

∫b

0 x

(^2) dx.
We first approximate the area under the curvey=x^2 between 0 andbbynrectangles of
equal widthh. If we take the value at the lower end of each subinterval (in the limit of an
infinite number of subintervals we could equally well have chosen the value at the upper
end) to give the height of the corresponding rectangle, then the area of thekth rectangle
will be (kh)^2 h=k^2 h^3. The total area is thus

A=

∑n−^1

k=0

k^2 h^3 =(h^3 )^16 n(n−1)(2n−1),

where we have used the expression for the sum of the squares of the natural numbers
derived in subsection 1.7.1. Nowh=b/nand so

A=

(

b^3

n^3

)

n

6

(n−1)(2n−1) =

b^3

6

(

1 −

1

n

)(

2 −

1

n

)

.

Asn→∞,A→b^3 /3, which is thus the valueIof the integral.

Some straightforward properties of definite integrals that are almost self-evident

are as follows:
∫b

a

0 dx=0,

∫a

a

f(x)dx=0, (2.23)

∫c

a

f(x)dx=

∫b

a

f(x)dx+

∫c

b

f(x)dx, (2.24)

∫b

a

[f(x)+g(x)]dx=

∫b

a

f(x)dx+

∫b

a

g(x)dx. (2.25)