To find the length of the curve represented by the vector function r(t) = (3t, 3cos(t), 3sin(t)), we need to compute the arc length using the formula:
L = ∫ab ||r'(t)|| dt
First, we differentiate the vector function r(t) with respect to t:
r'(t) = (3, -3sin(t), 3cos(t))
Next, we find the magnitude of the derivative:
||r'(t)|| = √(32 + (-3sin(t))2 + (3cos(t))2)
This simplifies to:
||r'(t)|| = √(9 + 9sin2(t) + 9cos2(t))
Using the Pythagorean identity sin2(t) + cos2(t) = 1, we get:
||r'(t)|| = √(9 + 9(1)) = √(18) = 3√2
The length of the curve from t = a to t = b is then:
L = ∫ab 3√2 dt
This results in:
L = 3√2 (b - a)
In summary, the length of the curve depends on the specific values of a and b that you choose for t.