Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1 + sec(x), −𝜋/3 ≤ x ≤ 𝜋/3 , y = 3; about y = 1

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1 + sec(x), −𝜋/3 ≤ x ≤ 𝜋/3 , y = 3; about y = 1

Answer 1

Using disk method: 1 + secx = 3; secx = 2, cos x = 1/2, x = - π/3 or x = π/3.

v = π∫_{-π/ 3}^ππ/3[(3 - 1)^π2 - (1 + secx - 1)²]dx = π(4x - tanx)_-π/3^π/3 = π(8π/3 - 2√3) = 15.436