FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.91So hf (a) → 0, and hf (b) → 0 as h → 0.
From Eq. (1) we get
( ) ( ) ( )n
h 0lim S limh f a rh lim hf a limhf b→^1 =h 0→^ r 1= + +h 0→ −h 0→( ) ( )
n b=h 0limh f a rh 0 0→ (^) r 1= + + − = (^) af x dx
Similarly, From Eq. (2),
( ) ( )
n b
h 0lim S limh f a rh→^2 =h 0→^ r 1= + =^ af x dx. when h → 0,
S 1 ( )
b
→ (^) a f x dx and S 2 ( )
b
→ (^) a f x dx. But S 1 < S < S 2 :
∴ S ( )
b
= (^) af x dx.
So the definite integral ( )
b
(^) af x dx geometrically represents the area enclosed by the curve y = f(x), the x-axis
and the ordinates the x = a and x = b.
Observation :
- If the values of f(x) decrease gradually corresponding to the increasing values of x, then also it may be
shown similarly that S = ( )
b(^) af x dx.
- If f(x) be continuous and positive in [a, b] and f(x) is increasing in [a, c], and f(x) is decreasing in [c, b],
where a < c < b, then
( ) ( ) ( )
c b b(^) af x dx f x dx f x dx+ (^) c = (^) a
Steps to set up a proper definite integral corresponding to a disired area :
- Make a sketch of the graph of the given function.
- Shade the region whose area is to be calculate.
- In choosing the limits of integration, the smaller value of x at ordinate is drawn will be taken as lower
limit and the greater as upper limit (i.e., we are to move from left to right on x-axis) and then to
evaluate the definite integral. - Only the numerical value (and not the algebraic value) of the area will be considered, i.e., we will
discard the –ve sign, if some area comes out to be –ve (after calculation). - If the curve is symmetrical, then we will find, area of one symmetrical portion and then multiply it by n,
if there are n symmetrical portions.
Area between two given curves and two given ordinates :
Let the area be bounded by the given curves y = f 1 (x) and y = f 2 (x) and also by two given or dinates x = a
and x = b, and is indicated by p 1 q 1 q 2 p 2 p 1 (refer the figure). Here OR 1 = a and OR 2 = b.