Find the volume of the solid bounded by the plane z=0 and the paraboloid z=1-x^2 –y^2
Accepted Solution
A:
Answer:π/2Step-by-step explanation:z = 1 − x² − y²We can integrate by first converting to cylindrical coordinates.z = 1 − r²At z = 0, r = 1. So the limits of integration are:0 ≤ r ≤ 10 ≤ θ ≤ 2π0 ≤ z ≤ 1 − r²The volume is:V = ∫ ∫ ∫ r dz dr dθV = ∫₀²ᴾ ∫₀¹ ∫₀ᶻ r dz dr dθV = ∫₀²ᴾ ∫₀¹ rz |₀ᶻ dr dθV = ∫₀²ᴾ ∫₀¹ r (1 − r²) dr dθV = ∫₀²ᴾ ∫₀¹ (r − r³) dr dθV = ∫₀²ᴾ (½ r² − ¼ r⁴) |₀¹ dθV = ∫₀²ᴾ ¼ dθV = ¼ θ |₀²ᴾV = π/2Another way we can look at this is by slicing the paraboloid into a stack of thin circular discs. Each disc has a volume of:dV = π r² dzWe know that r² = 1 - z, so:dV = π (1 - z) dzSo the volume is:V = ∫₀¹ π (1 - z) dzV = π (z - ½ z²) |₀¹V = π/2