Answer
$0$
Work Step by Step
We know that sine is an odd function and
$\sin (-x)=-\sin x$
Because of symmetry, the (negative) signed volume of the region below the $xy$-plane where $\pi \leq x \leq 2\pi$ cancels with the (positive) signed volume of the region above the xy-plane where $0 \leq x \leq \pi$.
Thus, the result is $0$.
