Yay uzunluğu formülü: $s = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx$
Birinci türev: $f'(x) = \frac{1}{x} - \frac{1}{4}x$
Türevin karesi: $[f'(x)]^2 = \left(\frac{1}{x} - \frac{1}{4}x\right)^2 = \frac{1}{x^2} - \frac{1}{2} + \frac{1}{16}x^2$
Kök içi ifade: $1 + [f'(x)]^2 = 1 + \frac{1}{x^2} - \frac{1}{2} + \frac{1}{16}x^2 = \frac{1}{x^2} + \frac{1}{2} + \frac{1}{16}x^2 = \left(\frac{1}{x} + \frac{1}{4}x\right)^2$

İntegral hesabı:$$s = \int_{2}^{4} \sqrt{\left(\frac{1}{x} + \frac{1}{4}x\right)^2} dx = \int_{2}^{4} \left(\frac{1}{x} + \frac{1}{4}x\right) dx$$$$s = \left[\ln x + \frac{1}{8}x^2\right]_{2}^{4} = \left(\ln 4 + \frac{16}{8}\right) - \left(\ln 2 + \frac{4}{8}\right)$$$$s = (\ln 4 + 2) - (\ln 2 + 0.5) = \ln\left(\frac{4}{2}\right) + 1.5 = \mathbf{\ln 2 + 1.5}$$