Let f(x) be a continuous function defied by
f(x) = 
Find the area bounded by the curve y = f(x), the x- axis and the line π y = 4 
(x– π )
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol . Since f(x) is everywhere continuous. Therefore,
f(x) = 
f(x) = f 
⇒ 
4 sin 
= 
{A (x – π ) 2 } + 2 = 4 sin
⇒ 
= 
+ 2 = 
⇒ A = (
–1)
Thus,
f(x) = 
For x ≤ , we have y = 4 sin
and for x > , y =
( 
–1) (x – π ) 2 + 2, which
represents a parabola having vertex at ( π , 2) and opens upward. The equation π y = 4 
(x – π ) represents a straight line passing through ( π , 0). The shaded region shown in fig. represents the desired region.
Now, required area
= Area of region OALO + Area of region LABML+ Area of region BMCB
= 
+ 
+ 
= – 8 
+ 
+ 
= – 8 
+ 2 π + 
– π + 
+ 3 π – –2 π
= – 4 
+ 8 + 2 π – +
× 
( 
–1)
= 4 
( 
–1) + (4
+1) sq. units.
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