3.90 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

3.5.7 GEOMETRICAL INTERPRETATION OF A DEFINITE INTEGRAL

( )

b

(^) a f x dx. Let f(x) be a function continuous in [a, b], where a and b are fixed finite numbers, (b > a). Let us
assume for the present f(x) is positive for a ≤ x ≤ b. As x increases from a to b, values of f(x) also increases.
In the figure the curve CD represents the function f(x), OA = a, OB = b, Ac = f(a) and BD = f(b).
Let S represent the area bounded by the curve y = f(x), the x-axis and the ordinates corresponding to x = a
and x = b.
Divide [a, b], i.e., part AB into n finite intervals each of length h so that nh = b – a or a + nh = b.
Let S 1 = sum of rectangles standing on AB and whose upper sides lie every where below the curve
y = f(x). and S 2 = sum of rectangles, whose upper sides lie above the curve y = f (x).
Now S 1 = hf (a) + hf (a + h) + .... + hf(a nh)+
( ) ( ) ( )
n
r 1
h f a rh hf a hf a nh

= + + − +

( ) ( ) ( ) ( )

n

1

=h f a rh hf a hf b ,.... 1 , + + − as a + nh = b

and S hf a h h a 2h ... hf a nh 2 = ( + )+ ( + )+ + ( + )

( )

n

r 1

h f a rh

=

= +

From the figure, it is now clear that S 1 < S < S 2. ...(2)

From nh = b – a, we get h=b an− , so as n → ∞, h → 0

Since f(a) and f(b) are finite numbers,