To find the length of the curve defined by the vector function r(t) = (r(t) cos(2t), sin(2t), 2 ln(cos(t))) over the interval [0, π/4], we need to use the formula for the arc length of a parametric curve.
The formula for the length L of a curve given by a vector function r(t) from t = a to t = b is:
L = ∫ab ||r'(t)|| dt
where ||r'(t)|| is the magnitude of the derivative of the vector function r(t).
First, we find the derivative r'(t) of the curve:
- x(t) = r(t) cos(2t)
- y(t) = sin(2t)
- z(t) = 2 ln(cos(t))
Now, we calculate the derivatives:
- x'(t) = r'(t) cos(2t) – 2r(t) sin(2t)
- y'(t) = 2 cos(2t)
- z'(t) = -2 tan(t)
Next, we find the magnitude of the derivative vector:
||r'(t)|| = √((x'(t))2 + (y'(t))2 + (z'(t))2)
Now, substitute the expressions for x'(t), y'(t), and z'(t) into the magnitude formula and simplify:
Next, we’ll set up the integral for the arc length from 0 to π/4:
L = ∫0π/4 ||r'(t)|| dt
Finally, computing this integral will give us the length of the curve over the specified interval. This calculation may require numerical methods or a calculator, depending on the complexity of ||r'(t)||.
This approach will help you find the length of the specified curve efficiently.