Find the moment about the x-axis of a wire of constant density that lies along the curve y=2(squarerootx) from x=0 to x=4

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!