Answer
$$S \approx 14.42359$$
Work Step by Step
$$\eqalign{
& y = \sin x,{\text{ }}0 \leqslant x \leqslant \pi \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\sin x} \right] \cr
& \frac{{dy}}{{dx}} = \cos x \cr
& {\text{Formula for surface area}} \cr
& S = 2\pi \int_a^b {r\left( x \right)\sqrt {1 + {{\left[ {f'\left( x \right)} \right]}^2}} } dx,{\text{ }}r\left( x \right) = f\left( x \right) \cr
& {\text{Then,}} \cr
& S = 2\pi \int_0^\pi {\left( {\sin x} \right)\sqrt {1 + {{\left( {\cos x} \right)}^2}} } dx \cr
& S = 2\pi \int_0^\pi {\left( {\sin x} \right)\sqrt {1 + {{\cos }^2}x} } dx \cr
& {\text{Integrate using a CAS or graphing utility}} \cr
& S \approx 14.42359 \cr} $$
