
15-6 Calculus – Differentiation and Integration
= −
[ ]
+
[ ]
= + =
cos cos
.
x x
0
2
4
p
p
p
2 2
Example 15.5 Find the area of the region bounded by the curves
y x
2
10= − and y x= +2 .
Solution: Given curves are y x
2
10= − and y x= +2 .
For simplification, we rewrite the curves as x y x y= − = −10 2
2 2
, ( ) .
The points of intersection are
10 2
10 4 4
2 4 6 0
1 3
2 2
2 2
2
− = −
⇒ − + = + −
⇒ − − =
⇒ = − =
y y
y y y
y y
y y
( )
and
Also note that the area to be computed is bounded above by the curve
y x
2
10= −
and bounded below
by y x= +2 .
The required area is given by
( ( ) ( )) ( ( ) )
( )
f y g y dy y y dy
y y y dy
a
b
−
∫
= − − −
∫
= − − − +
−
−
10 2
10 4 4
2 2
1
3
2 2
1
3
∫∫
= − +