Final answer:
To find the length of the curve defined by the parametric equations, we can use the arc length formula. The curve is defined by the parametric equations r(t) = 4t, t², (1/6)t³. To find the length, calculate dx/dt, dy/dt, dz/dt and substitute them into the arc length formula.
Step-by-step explanation:
To find the length of a curve defined by parametric equations, we can use the arc length formula. The formula is given by:
∫(√((dx/dt)² + (dy/dt)² + (dz/dt)²)) dt
In this case, the curve is defined by the parametric equations r(t) = 4t, t², (1/6)t³. To find the length of the curve, we need to calculate dx/dt, dy/dt, dz/dt and substitute them into the formula.
dx/dt = 4, dy/dt = 2t, dz/dt = (1/2)t²
Substituting these values into the arc length formula, we have:
∫(√(4²+ (2t)² + ((1/2)t²)²)) dt
To solve this integral, we can use integration techniques. Once the integral is solved, evaluate it from the lower limit 0 to the upper limit 1 to find the length of the curve.