Final answer:
To find the curve length, we calculate the magnitude of the derivative of r(t) and integrate it over the interval from t=0 to t=1. The provided formula in the question involving the cross product is not applicable in this case.
Step-by-step explanation:
To find the length of the curve r(t) = √2i + √2j + (1-t²)k from the point (0,0,1) to (√2,√2,0), we need to calculate the definite integral of the curve's derivative's magnitude over the given interval. We start by finding the derivative dr/dt, then integrate its magnitude from the start to the endpoint. The arc length formula for a smooth curve given by a vector function r(t) is usually expressed as ∫ |dr/dt| dt, where |dr/dt| represents the magnitude of the derivative of the vector function. In this case, however, the formula provided in the question, suggesting the use of the cross product to find arc length, is incorrect for this context and appears to be a red herring.
The derivative of r(t) is given by:
dr/dt = -2tk
Therefore, the magnitude of the derivative is:
|dr/dt| = |-2tk| = 2|t|
The arc length of the curve from t = 0 to t = 1 is calculated as:
∫0ⁱ |dr/dt| dt = ∫0ⁱ 2|t| dt
The calculation of this integral will give us the length of the curve from the initial to the final point.