The volume of the region described above (which lies in between the paraboloid) in the full question is given by Option C and Option A. because both.
We are given a region that lies between the paraboloid as:
z= 24 - x² - y; and
z = 2√(x² + y²)
Using cylindrical coordinates:
x = xcosθ, y = x sinθ, z = 2
dv = x dx dθ dx
z = 24 - t² ; and
z = 2e
⇒ 24 - t² = 2∈
∈² + 2t - 24 = 0
(∈ + 6) (∈-4) = 0
∈ = -6/ ∈ = 4
∈ ≠ -6
⇒ ∈ = 4
∈: 0 → 4, θ: 0 → 2π
⇒ Volume = [tex]\int_{0}^{2\pi}[/tex][tex]\int_{0}^{4}[/tex][tex]\int_{2t}^{24-E^2}[/tex]∈ dz dt dθ
= [tex]\int_{0}^{2\pi}[/tex]dθ [tex]\int_{0}^{4}[/tex] [tex][\mathrm{Z}]_{2E}^{24-E^2}[/tex]
= [tex][\mathrm{\theta}]_{\pi}^{2\pi}[/tex][tex]\int_{0}^{4}[/tex] ∈ - ∈³ - 2∈² dt
= 2π [tex]\int_{0}^{4}[/tex] 24∈ - ∈³ -2∈² dt
= 2π [192 - 64 - (128)/3]
= 2π [85.33]
= 536.16514
Learn more about paraboloid:
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Full Question:
Find the volume of the region E that lies between the paraboloid z= 24 - x² - y and the cone z = 2√(x² + y²)
A) 536.16514
B) 5361.651464
C) 536.16514
D) None of the above.
