whats the surface area of y= (x^3/6) + (1/(2x)) when x is between .5 and 1 and rotated about x-axis?

Answer 1

For an area there is no rotation.
A volume is obtained when there is a rotation.
Will go along area route:-
A = ∫ y.dx
A = ∫ (x³/6 + x / 2) dx lims 0.5 to 1 assumed
A = x^4 / 24 + x²/4
A = (1/24 + 1/4) - (1 / (24 x16) + 1 / 64)
A = 7 / 24 - (1/16).(1/24 + 1/4)
A = 7/24 - (1/16).(7/24)
A = (7/24).(1 - 1 / 16)
A = 7/24 x 15/16
A = 105/384 = 35 / 128 units²

Answer 2

A = int{2*pi*y*dx}; x from 0.5,1.0

A = 2pi*int{x^3/6 + (1/(2x))}

A = 2pi[x^4/24 + 1/2lnx + C]; x from 0.5,1.0

A = 0.77pi = 2.42 sq units