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