Final answer:
The student has asked to calculate the arc length of a 3-dimensional curve represented by a vector function over a specific interval. The solution involves taking the derivative of the vector function, finding its magnitude (the speed), and integrating this magnitude over the given interval.
Step-by-step explanation:
The student is asking about finding the arc length of a curve described by a vector function r(t) over a specific interval of the variable t. To find the arc length, one would typically integrate the magnitude of the derivative of the vector function (the speed function) over the given interval.
In this specific case, the curve is in 3-dimensional space, and we are provided with the vector function r(t) = (0, 6cos35t, 6sin35t) and the interval (3π/5) ≤ t ≤ (7π/10). To find the arc length As, one would first calculate the derivative of r(t), take its magnitude, and then integrate this magnitude from t = 3π/5 to t = 7π/10.
The formula for the arc length As is the integral of the speed function: As = ∫ ||r'(t)|| dt, where ||r'(t)|| represents the magnitude of the derivative of the vector function.