Question

Find the length of arc of the curve 𝑦 = 𝑥 3⁄2 /√3 from (0, 0) to (4, √8 / 3 )

Answer 1

If $x=4, y=8/\sqrt{3}$

$$f'(x)=(\dfrac{x^{3/2}}{\sqrt{3}})'=\dfrac{\sqrt{3}}{2}\sqrt{x}$$

$$L=\displaystyle\int_{0}^{4}\sqrt{1+(f'(x))^2}dx$$

$$=\displaystyle\int_{0}^{4}\sqrt{1+(\dfrac{\sqrt{3}}{2}\sqrt{x})^2}dx$$

$$=\dfrac{1}{2}\displaystyle\int_{0}^{4}\sqrt{4+3x}dx$$

$$=\dfrac{1}{2}[\dfrac{1}{3}(\dfrac{2}{3}(4+3x)^{3/2})]\begin{matrix} 4\\ 0 \end{matrix}$$

$$=\dfrac{1}{9}(64-8)=\dfrac{56}{9}$$