Chapter 7: Triple Integration
Section 7.5: Spherical Coordinates
Sketch the region above the cone z=x2+y2 but below the unit sphere that is centered at 0,0,1.
The cone and the sphere intersect in a circle whose radius is 1, whose center is at 0,0,1, and which lies in the plane z=1. This is established by solving the equations z=x2+y2 and x2+y2+z−12=1 simultaneously. Since these equations imply z2+z−12=1, the solutions are easily seen to be z=0 and z=1. Each generator of the cone therefore has length 2.
p1≔plot3d2 cosφ,θ=0..2 π,φ=0..π/4,coords=spherical:p2≔plot3dρ,θ,π/4,ρ=0..2, θ=0..2 π,coords=spherical:plots:-displayp1,p2,scaling=constrained,axes=frame,labels=x,y,z,tickmarks=3,3,3,orientation=−30,80,0,lightmodel=none
The sphere is given in spherical coordinates by ρ=2 cosφ, so the plot data-structure p1 houses the graph of the upper hemisphere. The cone is given in spherical coordinates by φ=π/4, so the plot data-structure p2 houses the graph of the cone.
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