Find the length of the curve r(t)=cos(4t) i+ sin(4t) j + 4ln(cos(t)) k 0<=t<=π/4?
Using the arc length formula I get to the integral of 16sec(t) evaluated at the given limits, but I can't seem to get the correct answer.
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Arc length = ∫√[(dx/dt)² + (dy/dt)² + (dz/dt)²] dt
= ∫√[16*sin²(4t) + 16*cos²(4t) + 16*tan²(t)] dt eval. from 0 to π/4
= ∫√(16 + 16*(sec²(t) - 1) dt eval. from 0 to π/4
= ∫4*sec(t) dt eval. from 0 to π/4
= 4*ln|sec(t) + tan(t)| eval. from 0 to π/4
= 4*ln(√2 + 1) - 4*ln(1)
= 4*ln(√2 + 1)