Finding length of curve

[samwhite]

Hi guys,
I'm a new member to the forum here.

So the question given is as follows:
y=(1/6)*(((x^2)+4)^(3/2))
Domain given is [0,2]
"Find the length of the curve, y, on the given domain using integrals."

So far I have been able to get it down to be the integral of (sqrt((x^4)+(4x^2)+1))
I cannot complete the square for this and I am stuck in terms of what to do next (or whether I need to go another way).

 

[Deleted member 4993]

Hi guys,
I'm a new member to the forum here.

So the question given is as follows:
y=(1/6)*(((x^2)+4)^(3/2))
Domain given is [0,2]
"Find the length of the curve, y, on the given domain using integrals."

So far I have been able to get it down to be the integral of (sqrt((x^4)+(4x^2)+1))
I cannot complete the square for this and I am stuck in terms of what to do next (or whether I need to go another way).

x⁴ + 4x² + 1 = x⁴ + 4x² + 4 - 3 = (x² + 2)² - 3

but

y' = (1/6)*(3/2)[((x^2)+4)^(1/2)]*2*x = (1/2)[((x^2)+4)^(1/2)]*x

1 + (y')² = (1/4)[x^2 + 4]*x² + 1 = (1/4)[x⁴ + 4x² + 4] = [1/2 * (x² + 2)]² = \(\displaystyle \left [\dfrac{x^2}{2} + 1\right ]^2\)