Question

The length of the curve $y=\frac{2}{3}{x}^{3/2}$ between $x=0$ and $x=1$ is

• A. $0.27$
• B. $0.67$
• C. $1$
• D. $1.22$

Answer 1

The correct option is D $1.22$

Given, curve is $y=\frac{2}{3}{x}^{3/2}$
We know that
Length of a curve $={\int }_{x_1}^{x_2}\left(\sqrt{1+(\frac{dy}{dx}{)}^2}\right)dx$
$\therefore$ Length of the given curve between $x=0$ and $x=1$
$={\int }_0^1\left(\sqrt{1+x}\right)dx(\because \frac{dy}{dx}=x^{1/2})$
$={\left[\frac{(1+x{)}^{3/2}}{3/2}\right]}_0^1$
$=\frac{2}{3}[2^{3/2}-1]\simeq 1.219\simeq 1.22$