Which of the following integrals represents the volume of the solid obtained by rotating the region bounded by the curves $y=\sin(x)$ and $y=0,$ with $0 \le x \le \pi$ about the line $y=1?$

A. $\displaystyle \pi \int_{0}^{\,\pi} {[(1)-(1-\sin(x))]^2}\,dx$
B. $\displaystyle \pi \int_{0}^{\,1} {[1^2-(1-\sin(x))^2]}\,dx$
C. $\displaystyle \pi \int_{0}^{\,\pi} {[1^2-(1-\sin(x))^2]}\,dx$
D. $\displaystyle \pi \int_{0}^{\,1} {[1-\sin^2(x)]}\,dx$
E. $\displaystyle \pi \int_{0}^{\,1} {[(1)-(1-\sin(x))]^2}\,dx$
F. $\displaystyle \pi \int_{0}^{\,\pi} {[1-\sin^2(x)]}\,dx$