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Multi-variable Calc question Use cylindrical coordinates to evaluate the triple integral?
Use cylindrical coordinates to evaluate the triple integral sqrt(x^(2)+y^(2)) dV , where E is the solid bounded by the circular paraboloid z = 16-4(x^(2)+y^(2) and the xy - plane.
1 Answer
- kbLv 78 years agoFavorite Answer
Using cylindrical coordinates:
z = 16 - 4(x^2 + y^2) = 16 - 4r^2.
This crosses the xy-plane (z = 0) when 0 = 16 - 4r^2 ==> r = 2.
So, ∫∫∫ √(x^2 + y^2) dV
= ∫(θ = 0 to 2π) ∫(r = 0 to 2) ∫(z = 0 to 16 - 4r^2) r * (r dz dr dθ), via cylindrical coordinates
= 2π ∫(r = 0 to 2) r^2 (16 - 4r^2) dr
= 2π ∫(r = 0 to 2) (16r^2 - 4r^4) dr
= 2π (16r^3/3 - 4r^5/5) {for r = 0 to 2}
= 512π/15.
I hope this helps!
