Integrating with shells is the easier method.
V = 2π ∫₁³ x (√x + 3x) dx
That is, at various values of x in the interval [1, 3], we take n shells of radius x, height y = √x + 3x, and thickness ∆x so that each shell contributes a volume of 2π x (√x + 3x) ∆x. We then let n → ∞ so that ∆x → dx and sum all of the volumes by integrating.
To compute the integral, just expand the integrand:
V = 2π ∫₁³ (x ³ʹ² + 3x ²) dx
V = 2π (2/5 x ⁵ʹ² + x ³) |₁³
V = 2π ((2/5 × 3⁵ʹ² + 3³) - (2/5 × 1⁵ʹ² + 1³))
V = 4π/5 (9√3 + 64)
