Arc Length

The length of a differentiable function \( y = f(x) \) from \( a \) to \( b \) is given by \[ L = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx \]

Find the exact length of the curve:

1. \( y = x^{3/2} \)   from \( x = 0 \) to \( x = 4 \) Solution ↗
2. \(\displaystyle f(x) = \ln(\sec x), \quad 0 \leq x \leq \frac{\pi}{4} \) Solution ↗
3. \(\displaystyle y = \frac{x^3}{3} + \frac{1}{4x} \)   from \( x = 1 \) to \( x = 2 \) Solution ↗
4. \(\displaystyle y = 2\ln\!\left(\cos \tfrac{1}{2}x\right), \quad 0 \leq x \leq \frac{\pi}{3} \) Solution ↗
5. \(\displaystyle y = \frac{1}{4}x^2 - \frac{1}{2}\ln x, \quad 1 \leq x \leq 6 \) Solution ↗