Let #dl# be the element of length of the curve between the points #x# and #x+dx# with #q <= x <= u#:
#dl = sqrt(dx^2+dy^2) = sqrt(dx^2 + (f'(x)dx)^2) =dx sqrt(1+(f'(x))^2)#
so that the length of the curve is:
#L=int_q^u sqrt(1+(f'(x))^2) dx#
The element of surface generated by the rotation of #dl# around the #x# axis is then:
#dS = 2pif(x)dl = 2pi f(x) sqrt(1+(f'(x))^2)#
and the surface is:
#S_f = 2pi int_q^u f(x) sqrt(1+(f'(x))^2) dx#
Consider now the curve generated by the rotation of:
#g(x) = f(x) + c#
Clearly #g'(x) = f'(x)#, so:
#S_g = 2pi int_q^u g(x) sqrt(1+(g'(x))^2) dx#
#S_g = 2pi int_q^u (f(x)+c) sqrt(1+(f'(x))^2) dx#
and as the integral is linear:
#S_g = 2pi int_q^u f(x) sqrt(1+(f'(x))^2) dx + 2pic int_q^u sqrt(1+(f'(x))^2) dx#
that is:
#S_g = S_f + 2picL#
