Cauchy
(1789-1857)
360
MATHEMATICS
Fig  8 2
This area is called the elementary area which is located at an arbitrary position
within the region which is specified by some value of x between a and b We can think
of the total area A of the region between x-axis, ordinates x = a, x = b and the curve
y = f (x) as the result of adding up the elementary areas of thin strips across the region
PQRSP Symbolically, we express
A = 
A
( )
b
b
b
a
a
a
d
ydx
f x dx
=
=
∫
∫
∫
The area A of the region bounded by
the curve x = g (y), y-axis and the lines y  =  c,
y = d is given by
A = 
( )
d
d
c
c
xdy
g y dy
=
∫
∫
Here, we consider horizontal strips as shown in
the Fig 8