Question

The region in the first quadrant bounded by the curve 𝑦 = cos 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑦 = 1 𝑎𝑛𝑑 𝑥 =1/2𝜋 𝑖𝑠 𝑟𝑒𝑣𝑜𝑙𝑣𝑒𝑑 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑥 =1/2𝜋. Find the volume of the solid generated.

Answer 1

using cylinders of thickness dx, we have
v = ∫[0,π/2] 2πrh dx
where r = π/2 - x and h = 1-cosx
v = ∫[0,π/2] 2π(π/2 - x)(1-cosx) dx = π/4 (π^2 - 8) ≈ 1.4684