Use polar coordinates to find the volume of the solid

Question

Use polar coordinates to find the volume of the solid below the paraboloid z=75−3x^2−3y^2 and above the xy-plane.

Answer 1

Volume V = ∫∫Rzdxdy.

We will use polar coordinates: x = rcosθ, y = rsinθ.

Then z = 75 - 3r²; R: z = 0, 75 - 3r² = 0, r² = 25, r = 5. region r ≤ 5, 0 ≤ θ ≤ 2π; dxdy = rdrdθ. Then

V = ∫∫R(75 - 3r²)rdrdθ = ∫₀^2π∫₀⁵(75r - 3r³)drdθ = (75r²/2 - 3r⁴/4)₀⁵·θ₀^2π =(75·25/2 - 3·625/4)·2π = 1875π/2 ≈2945.24