Question

Find the arc length parameter along the given curve from the point where t0 by evaluating the integral s(t). Then find the length of the indicated portion of the curve r(t) i j k, where 0 ≤ t.

Answer 1

The arc length parameter along the given curve from the point where
`\( t = t_0 \) is given by \( s(t) \).`To find the length of the indicated portion of the curve
`\( r(t) = i \hat{} + j \hat{} + k \hat{} \) where \( 0 \leq t \)`, evaluate the integral of
`\( s(t) \)` over the specified interval.

Step-by-step explanation:

To determine the arc length parameter along a curve, we utilize the formula
`\( s(t) = \int \sqrt{\left((dx)/(dt)\right)^2 + \left(\frac{dy}`
`{dt}\right)^2 + \left((dz)/(dt)\right)^2} \, dt \).` This represents the integral of the magnitude of the velocity vector over the given interval.

In the context of the given curve
`\( r(t) = i \hat{} + j \hat{} + k \hat{} \), we compute \( s(t) \)` by evaluating the integral mentioned above. This yields a function
`\( s(t) \)`that provides the arc length parameter along the curve.

The final answer instructs us to find the length of the indicated portion of the curve by integrating
`\( s(t) \) over the specified interval \( 0 \leq t \).`This integral operation essentially calculates the arc length of the curve within the given range, providing a quantitative measure of the distance along the curve from the initial point
`\( t_0 \).`