Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.79

is denoted symbolically by

a

b

f(x)dx i.e.,

a n

(^) bf(x)dx limh f(a rh),b a,b a nh.=h 0→^ r 1= + > − =
Note: (i) a is called as lower limit, while b is known as upper limit.
(ii) If a = 0, then
b n
(^) af(x)dx limh f (rh) here nh b=h 0→^ r 1= =
(iii) If a = 0, b = 1, then
(^1) n
0 f(x)dx limh f (rh) where nh 1=h 1→^ r 1= =
(iv) if a > b, then
b b
a a
f(x)dx f(x)dx.= −
(v) If a = b then
b
a
f(x)dx 0=
Example 163:Evaluate the following definite integral from definition
b
a
2dx.
Here, f(x) = 2 (a constant) ∴ f(a + rh) = 2, nh = b – a.
Now from
b n
a^ f(x)dx limh f (a+rh) we find=h 0→^ r 1=
b n
(^) a2dx limh 2 = limh.2n lim 2 nh=h 0→^ r 1= h 0→ =h 0→
=lim2(b -a), as nh = b-a = 2(b-a)h 0→
Example 164: Evaluate from the first principle the value of
b
a
xdx.
Here, f(x) = x ∴ f(a + rh) = a + rh, nh = b – a
Now from
b n
(^) af(x)dx limh f(a rh),=h 0→^ r 1= + we get

{ }

b n

(^) axdx limh (a rh) limh na h(1 2 3 ... n)=h 0→^ r 1= + =h 0→ + + + + +
h 0
limh na h.n(n 1) ;as1 2 ... n n(n 1)
→ 2 2

= + +^ + + + = +

h 0

lim a(nh) (nh)(nh h)

→ 2

=^ + +^