Question

Let D be the region in R^3 that lies inside the cone z=sqr(x^2+y^2) above the plane z=1 and below the hemisphere z=sqr(4-x^2-y^2). (a) sketch the region D in R^3. (b) express the volume of D as a sum of triple integrals, using cylindrical coordinates.

Accepted Answer

In spherical coordinates ρ, ϕ, θ where 0≤ ϕ ≤ π and 0≤ θ
≤ 2π given surfaces may by described by equations:

·        cone z=sqr(x^2+y^2):  ϕ = π/4

·        sphere z=sqr(4-x^2-y^2): ρ = 2

·        plane z=1: from equality z = ρ cos ϕ => ρ =
z/cos ϕ and for z = 1 ρ = 1/cos ϕ

So

V\ =\ \iiint_{D}{dV\ =\ }\ \iiint_{D}{dV\ =\ }\
\iiint_{D}{ho^2sin\varphi dho d\varphi
d	heta}=\int_{0}^{2\pi}{\int_{0}^{\pi/4}{\int_{1/cos
\varphi}^{2}dho d\varphi}d	heta}

Question #218848

Expert's answer

Answer in progress...