10
Fig 8 11
Y
O
A
y
x
=
Y'
B
M
(4�4)
,
X
X'
(4
2 0)
,
APPLICATION OF INTEGRALS         365
O
F (
o)
ae,
B
Y
Y′
B'
S
R
X
X′
x
ae
=
Again, the area of the region BMAB
= 
4 2
4
ydx
∫
= 
4 2
2
4
32
−x dx
∫
= 
4 2
2
–1
4
1
1
32
32
sin
2
2
4 2
x
x
x


−
+
×
×




= 
–1
–1
1
1
4
1
1
4 2
0
32
sin 1
32
16
32
sin
2
2
2
2
2




×
+
×
×
−
−
+
×
×








= 8 π – (8 + 4π) = 4π – 8 (4)
Adding (3) and (4), we get, the required area = 4π Example 5 Find the area bounded by the ellipse 
2
2
2
2
1
x
y
a
b
+
=  and the ordinates x = 0
and x = ae, where, b2 = a2 (1 – e2) and e < 1