The area bounded by the curve y=sin x between x=0 and x= 2π is :

by symmetry

area from $0 → \frac{π}{2}$

= area from $\frac{π}{2}→π$

= area from $π→\frac{3π}{2}$

= area from $\frac{3π}{2}→2π$

so area = 4 area from $0 → \frac{π}{2}$

$=4\int\limits_{0}^{π/2}\sin xdx$

$=4[-\cos x]_{0}^{π/2}$

= 4 sq. units