Final answer:
To find the length of the curve, use the formula for arc length and calculate the integral of the derivative components squared. Apply this formula to the given curve r(t) = 5t, 3 cos(t), 3 sin(t), where t ranges from -3 to 3. Integrate the expression and simplify to find the length of the curve.
Step-by-step explanation:
The length of a curve can be found using the formula for arc length:
L = ∫(a,b)√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
For the given curve r(t) = 5t, 3 cos(t), 3 sin(t), and t ranges from -3 to 3, we can calculate the derivatives:
dr/dt = 5, -3 sin(t), 3 cos(t)
Using these derivatives, we can calculate the integrand:
√(dx/dt)² + (dy/dt)² + (dz/dt)² = √(5)² + (-3 sin(t))² + (3 cos(t))²
Next, we integrate this expression over the given range of t from -3 to 3:
L = ∫(-3,3) √(5)² + (-3 sin(t))² + (3 cos(t))² dt
Simplifying and evaluating this definite integral will give us the length of the curve.