Find the arc length of $f(x)=x^{\frac{3}{2}}$ from $x=0$ to $x=1$.

The arc length of a function on the interval $(a,b)$ is computed by the formula
\[L=\int_{a}^{b}\sqrt{1+(f'(x))^2}dx.\]
Hence
\[L=\int_{0}^{1}\sqrt{1+(\frac{3}{2}x^{\frac{1}{2}})^2}dx=\int_{0}^{1}\sqrt{1+\frac{9}{4}x}dx=\int_{0}^{1}(1+\frac{9}{4}x)^{\frac{1}{2}}dx\]
\[= \frac{(1+\frac{9}{4}x)^{\frac{3}{2}}}{\frac{9}{4}\times \frac{3}{2}}|_0^1=\frac{8}{27}\left ((1+\frac{9}{4})^{\frac{3}{2}}-1\right)\]
\[=\frac{8}{27}\left( (\frac{13}{4})^{\frac{3}{2}}-1\right).\]