Surface area of revolution
y=(x^3)+3 about the x-axis from (0,1)
a = ∫[0,1] 2π y ds
where ds = √(1+(y')^2)
a = 2π∫[0,1] (x^3+3)√(1+9x^4) dx
The ∫(x^3)√(1+9x^4) dx
part is easy, but the
∫3√(1+9x^4) dx
cannot be done with elementary functions.