Page 49 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
Figure 4.1
Y

y=f(x)

f(a+h) f(a+2h) f(a+3h)
f(a+n-1h)
f(a) f(b)

0 a a+h a+2h a+3h a((n-1)h b X

The sum of these areas:

(


a

h
h

A hf ( ) hf (a h ) hf a  2 )  hf (a 3 )   hf ( a n 1 )
h

h

h
h

)
a
f
 h [ ( )  ( f a h  ( f a  2 )  ( f a 3 )    hf ( a n 1 )]
This area is close to the area of the region bounded by the curve y = f(x), x-axis and the ordinates
x = a, x = b.
If n increases, the number of rectangles will increases and the width of rectangles will decrease.
Did u know? A will give closer approximation of the area enclosed by the curve y = f(x),
x-axis and the coordinates x = a, x = b.
Thus the area of region bounded by curve y = f(x), x-axis and the ordinates x = a, x = b is
a
f
h

h
lim [ ( )  ( f a h  ( f a  2 )  ( f a  3 )   ( f a n  1 ]
)
h

h
n

b a
where h
n
f
 lim [ ( )  ( f a h  ( f a  2 )  ( f a  3 )   ( f a n  1 ]
h
a
h

)
h
h

h  0

where nh b a [ as n  ,h 0]


Notes This area is also the limiting value of any area which is among that of the rectangles
beneath the curve and that of the rectangles over the curve.
!
Caution For the sake of ease, we shall consider rectangles with height identical to that of
the curve at the left hand edge of every subinterval.
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