Evaluate the following iterated integral using cylindrical coordinates.

Question

Evaluate the following iterated integral using cylindrical coordinates.

∫_-4⁴ ∫₀^{√16-x^2} ∫₀^y y dz dy dx

Answer 1

-To convert cartesian coordinates to cylindrical we have:

z = z

y = r.sin(θ)

x = r.cos(θ)

-To convert variable of integration from cartesian coordinates to cylindrical we have:

dzdydx = rdrdθdz

-To convert integral limits from cartesian coordinates to cylindrical we have:

z = [0,y] → z = [0,r.sin()]

y = [0,√(16-x²)], r.sin(θ) = √(16-r²cos²(θ)) → r = ± 4, -4 is not acceptable → r=[0, 4]

x = [0,4] → θ = [0,π]

∫_-4⁴ ∫₀^{√16-x^2} ∫₀^y y dzdydx = ∫_-4⁴( ∫₀^{√16-x^2} (∫₀^y y dz) dy) dx =

∫₀^π( ∫₀⁴ (∫₀^r.sin(θ) r.sin(θ) dz) r.dr) dθ = ∫₀^π( ∫₀⁴ [r.sin(θ).z]₀^r.sin(θ) r.dr) dθ =

∫₀^π( ∫₀⁴ r².sin²(θ) .r.dr) dθ = ∫₀^π [r⁴.sin²(θ)/4]₀⁴ dθ =

∫₀^π 64.sin²(θ) dθ = ∫₀^π 32.(1-cos(2θ)) dθ] = 32.[(π-0.5.sin(2θ))]₀^π =

32.((π-0)-(0-0)) =

32π (Answer)