Final answer:
To find the length of the parametric curve r(t) = (6t, t², 19t³), we use the arc length formula, integrate the square root of the sum of the squares of the derivatives of each component of r(t), from t=0 to t=1.
Step-by-step explanation:
Finding the Length of the Curve
To find the length of the curve r(t) = (6t, t², 19t³) from t = 0 to t = 1, we use the formula for the arc length of a parametric curve in three dimensions, which is:
L = ∫_0^1 √((dx/dt)² + (dy/dt)² + (dz/dt)² ) dt
Taking the derivatives of each component of the position vector r(t), we find:
- dx/dt = d/dt (6t) = 6
- dy/dt = d/dt (t²) = 2t
- dz/dt = d/dt (19t³) = 57t²
Substituting these derivatives into the arc length formula, we get:
L = ∫_0^1 √(6² + (2t)² + (57t²)² ) dt
Integrating this expression from t=0 to t=1 will give us the total arc length of the curve.