Did you man y = sqrt (x)?
The moment of inertia is
I = ∫r²dm
In this case, the integration is along the x-axis from 0 to X (y is not defined for x < 0). The moment must be limited to finite distance, since it will go to ∞ if x --> ∞
dm = ρ*ds, where ρ is the linear density.
r = y = √x; r² = x
ds = √[1 + y'²]*dx where y' = dy/dx = (1/2)*x^-(1/2)
ds = √[1 + (1/4)*x^(-1)]*dx
Then
I =ρ*∫x*√[1 + 1/(4*x)]*dx
I =ρ/2*∫√[4*x² + x)]*dx
ANALYZING THE SOLUTION MIGHT JUST HELP YOU TO ANSWER THE PROBLEM ON YOUR OWN!
