Find the Length of the Curve: r(t) = (cos(7t), sin(7t), 7 ln(cos(t))), for 0 ≤ t ≤ π/4

To find the length of the given curve, we can use the formula for the arc length of a parametric curve defined by the vector function r(t).

The arc length L from t = a to t = b is given by:

L = ∫ab ||r'(t)|| dt

Here, r(t) = (cos(7t), sin(7t), 7 ln(cos(t))). We need to first calculate the derivative r'(t).

Calculating the components of r'(t):

• x(t) = cos(7t) → x'(t) = -7sin(7t)
• y(t) = sin(7t) → y'(t) = 7cos(7t)
• z(t) = 7 ln(cos(t)) → z'(t) = -7tan(t)

Next, we find the magnitude of r'(t):

||r'(t)|| = √((-7sin(7t))2 + (7cos(7t))2 + (-7tan(t))2)
        = √(49(sin²(7t) + cos²(7t) + tan²(t)))
        = 7√(1 + tan²(t))
        = 7sec(t)

Now, we integrate to find the length:

L = ∫0π/4 7sec(t) dt

The integral of sec(t) is ln|sec(t) + tan(t)|. Thus, we get:

L = 7[ln|sec(t) + tan(t)|]0π/4

Calculating the limits:

• At t = π/4: sec(π/4) + tan(π/4) = √2 + 1
• At t = 0: sec(0) + tan(0) = 1 + 0 = 1

Now substituting back into our integral:

L = 7[ln(√2 + 1) - ln(1)]
  = 7ln(√2 + 1)

Thus, the length of the curve from t = 0 to t = π/4 is:

L = 7ln(√2 + 1).