5.5.8 Let E be the region bounded cone z =z = x² + y² + z². Provide an answer accurate to at least 4 significant digits.Find the volume of E.0.8-Z0.60.40.2-0-0.4Triple IntegralSpherical CoordinatesCutout of sphere is for visual purposes-0.20X0.22 ⋅ (x² + y²) and the sphere.0.4 -0.4-0.200.20.4Note: The graph is an example. The scale and equation parameters may not be thesame for your particular problem. Round your answer to 4 decimal places.Hint: Convert from rectangular to spherical coordinate system.

Given: Cone: z=2x2+y2 Sphere: z=x2+y2+z2 To find: The volume of the bounded surface using spherical coordinates. Cartesian coordinates to Spherical coordinates: x = ρ sin ϕ cos θy = ρ sin ϕ sin θz = ρ cos ϕ where, ρ is the distance from the origin to any point x,y,z on the surface. ϕ is the angle measured downward from positive z axis. θ is the usual polar angle. Integration Formulae: ∫sin t dt = -cos t + c∫xn dx = xn+1n+1+c where, c is the integration constant. First, we can find the cylindrical coordinates. x=r cos θy=r sin θz=zdxdydz=rdrdθdz where r=x2+y2 Find the limit of r: Cone: z=2r2 Sphere: z=r2+z2 Equating both equations: z-z2 = z22z = z22+z2z = 32z21 = 32zz = 23 Hence, the limit of z is 0 to 23Change the coordinates into spherical coordinates: x = ρ sin ϕ cos θy = ρ sin ϕ sin θz = ρ cos ϕdxdydz=ρ2 sin ϕ dρ dϕ dθ where, ρ=r2+z2 and ϕ=cos-1zz2+r2 ρ=r2+z2=z-z2+z2=z=23 ϕ= cos-1zz= cos-1z= cos-1ρ The value of ρ is 0 to 23. The value of ϕ is 0 to cos-1ρ. The value of θ is 0 to 2π. Volume of the region: V=∫02π∫023∫0cos-1ρρ2 sin ϕ dϕ dρ dθ=∫02π∫023ρ2∫0cos-1ρsin ϕ dϕ dρ dθ=∫02π∫023ρ2-cos ϕ0cos-1ρdρ dθ=∫02π∫023ρ2-cos cos-1ρ+cos 0 dρ dθ=∫02π∫0231-ρρ2dρ dθ=∫02π∫023ρ2-ρ3 dρ dθ=∫02πρ33-ρ44023 dθ=∫02π132332-14232dθ=132332-192π≈0.0703325732π≈0.441912592≈0.4419 Hence, the approximate volume of the given surface is 0.4419 cubic units.