Answer
$\dfrac{8}{27}(10\sqrt {10}-1)$
Work Step by Step
We know that the formula to calculate the arc length is defined as: $L=\int_m^n \sqrt {1+[f'(x)]^2} dx$
This implies that $L=\int_0^4 \sqrt {1+[(\dfrac{3}{2})x^{(1/2)})]^2}dx=\int_0^4 \sqrt {1+(\dfrac{9}{4})x} dx$
$\implies \int_0^4 (1+(\dfrac{9}{4})x)^{1/2} dx=(\dfrac{2}{3})(\dfrac{4}{9}) [1+(\dfrac{9}{4}) (x)^{(3/2)}]_0^4=\dfrac{8}{27}(10\sqrt {10}-1)$
