3.80 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS
Calculus
h 0
limh a(b a) (b a)(b a h)
→ 2
=^ − + − − +^
( ) ( ) ( )
b a^2 b a
=a b a− + − 2 = b a a−^ + − 2
( ) ( ) (^22 )
b a b a^1 b a
2 2
= − + + = −
Example 165:By the method of summation, find the value of
1
0
(3x 5)dx+
Here, f(x) = (3x+5); a = 0, b = 1;
f(a + rh) = 3 (a + rh) + 5 = 5 +3rh, nh = b – a = 1 – 0 = 1
(^1) n n
∴ 0 (3x 5)dx limh f(a rh) limh (5 3rh)+ =h 0→ (^) r 1= + =h 0→ (^) r 1= +
=h 0limh 5n 3h(1 2 3 ... n) limh 5n 3h.→ { + + + + + }=h 0→^ + n(n 1) 2 +^
h 0 h 0
lim 5.nh (nh)(nh h) lim 5.1 .1.(1 h)^33
→ 2 → 2
= + =^ + +^
5.1 .1.1 5^3 3 10 3 13.
2 2 2 2
= + = + = + =
SELF EXAMINATION QUESTIONS
1.
b
a
dx [Ans. b – a] 2.
b
a
10dx. [Ans. 10(b–a)
3.
b 3
a
x dx [Ans.^1 (b a )^44
4 − ] 4.
1
0
dx [Ans. 1]
5.
1
0
xdx [Ans.^1
2 ] 6.
(^12)
0
x dx [Ans.^1
3
7.
(^22)
1
5x dx [Ans.^35
3 ] 8.
1
0
(2x 5)dx+ (Ans. 6]