Let f(x) be a non-negative differentiable function satisfying the differential equation 
= x for x ≤ 
and the differential equation y 
= x for x > 
. Find the area bounded by the curve y = f(x), the x- axis and the ordinates at x = 0 and x = 2.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol Since f(x) is differentable at x = 
. Therefore,
(LHD at x = 
) = (RHD at x = 
)
⇒ 
= 
⇒ 
x = 
⇒ 
= 
[ f(x) is continuous at x = 
]
⇒ f ( 
) = 1
Now, 
= x for x ≤ 
⇒ dy = x dx
⇒ y = 
+ C 1
But, y = 1 when x = 
. Therefore,
1 = 
+ C 1 ⇒ C 1 = 0
∴ y = 
for x ≤ 
Also, y 
= x for x > 
⇒ y dy = x dx for x > 
⇒ 
= 
+ C 2
But, y = 1 when x = 
. Therefore,
= 1 + C 2 ⇒ C 2 = – 
∴ 
= 
– 
for x > 
⇒ y 2 = x 2 –1 for x > 
Thus, we have,
f(x) = 
The graph of the curve y = f(x) is shown in fig. We have to find the area of the shaded region.
Now,
Required area
= 
+ 
= 
+ 
= 
+ 
+ 
log (2 + 
) – 
– 
log (1 + 
)
= 
sq. units.
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems