Surface area- calculus help please?

Question

find the area of the surface generate by revolving about the x-as the arc of the curve y = x^1/2 from [1,4]

thank you

Answer 1

A = 2π ∫ f(x) √(1 + f'(x)^2) dx

y = x^1/2

y' = (1/2) x^-1/2

A = 2π ∫ (x^1/2) √(1 + 1/(4x)) dx

A = 2π ∫ √( x + 1/4 ) dx

= 2π *(2/3) (x + 1/4)^(3/2)

= 4/3 π (x + 1/4)^(3/2) .. . . evaluated from 1 to 4

= (4/3)π (4.25^1.5 - 1.25^1.5)