Summary Mathematics for Engineering, ISBN: 9781136075490 Engineering maths

Applications of
Integrals
The space occupied by the curve along with the axis, under the given
condition is called area of bounded region.
(i) The area bounded by the curve y = F ( x ) above the X-axis and
between the lines x = a , x = b is given by
Y
y = F(x)

y
x=a x=b
X′ dx X
O
Y′
b b
∫ a y dx = ∫ a F( x ) dx
(ii) If the curve between the lines x = a , x = b lies below the X-axis,
then the required area is given by
Y

dx
X′ X
O
x = a −y x=b


Y′ y = F(x)
b b b
|∫ ( − y ) dx| =|− ∫ a y dx| =|− ∫ a F( x ) dx|
a

, (iii) The area bounded by the curve x = F ( y ) right to the Y -axis and
between the lines y = c, y = d is given by
d d
∫c x dy = ∫c F ( y ) dy

Y

y=d

x
dy
x = F(y)

y=c
X' X
O
Y'

(iv) If the curve between the lines y = c, y = d left to the Y -axis,
then the area is given by
Y

y=d

x = F(y) –x
dy


y=c
X' X
Y'
d d d
|∫ ( − x ) dy| =|− ∫c x dy| =|− ∫c F ( y ) dy|
c

(v) Area bounded by two curves y = F ( x ) and y = G ( x ) between
x = a and x = b is given by
Y
y = F(x)



y = G(x)



X′ O x=a x=b X
Y′
b
∫a { F ( x ) − G( x )} dx