Find the arc length of the given function/curve on the given interval.
y=ln(x-sqrt(x^(2)-1)); x ϵ [1, sqrt(2)]
To find the arc length of a function or curve, we can use the formula:
L = ∫[a, b] sqrt(1 + (f'(x))^2) dx
Where,
L: Arc length
∫: Integration
[a, b]: Interval
f(x): Function or curve
In this case, the given function is y = ln(x - sqrt(x^2 - 1)) and the interval is x ϵ [1, √2].
Let's first find the derivative of the function f(x) = ln(x - sqrt(x^2 - 1)) to calculate (f'(x))^2.
f'(x) = 1 / (x - sqrt(x^2 - 1)) * (1 - (2x)/2sqrt(x^2 - 1))
Simplify:
f'(x) = 1 / (x - sqrt(x^2 - 1)) * (1 - x / sqrt(x^2 - 1))
Now we can substitute this derivative into the arc length formula:
L = ∫[1, √2] sqrt(1 + (1 / (x - sqrt(x^2 - 1)) * (1 - x / sqrt(x^2 - 1)))^2) dx
To solve this integral, it is best to use numerical methods or a computer program like Python or Wolfram Alpha to calculate the result.