Answer
$6$
Work Step by Step
Step 1. The limit of the Riemann sum leads to a definite integral, and from the expression given in the exercise, we can identify that $f(x)=x(x^2-1)^{1/3}$ and the interval is $[1,3]$.
Step 2. We can write the integral as
$\int_1^3f(x)dx=\int_1^3x(x^2-1)^{1/3}dx$
Step 3. Using substitution, let $u=x^2-1$ and we have $du=2x\ dx$ with $x\to1, u\to0$ and $x\to3, u\to8$
Step 4. We have
$\int_1^3f(x)dx=\frac{1}{2}\int_0^8(u)^{1/3}du=\frac{3}{8}u^{4/3}|_0^8=\frac{3}{8}(8^{4/3})=6$
