7f501bf4-e4bc-4b72-b446-df919244ddd2 8 4 5 false false false 0 3 decimal

Volume from https://smath.com/en-US/forum/topic/nGKRV-last/Find-the-formula-for-the-volume-of-a-solid#msg0

Unit vectors of cylinders

c 13 sqrt / :

U 100 c c c c - 010 c c c - c 001 c c - c c 37 mat :

k u U k col :

k 17 range :

Normal in spherics

n θ sin φ cos * θ sin φ sin * θ cos 31 mat :

Position vector of the surface

r R N k el n k u † norme : max / :

θ φ S r 3 identity * n * :

Radius of the cylinders

R 4 :

CreateMesh mathcad :

Surface plot and Volume value

S 0 2 π * 0 π 50 50 CreateMesh

θ φ f r 3^3 / θ sin * abs eval :

V θ φ f θ 0 π int φ 02 π * int :

Unfortunatelly, SMath can't eval this integral.

Numerical procedure

2 f x.1 x.2 y.1 y.2 n.x n.y DInt a b n RK M 011 mat a b n x y D dn_AdamsMoulton : M M rows 2 el eval 21 line : y u D x v D x y f 11 mat : x.1 x.2 n.x RK 21 line : y.1 y.2 n.y RK 31 line :

V θ φ f 02 π * 0 π 10001000 DInt :

V 274.0122

Circumscript and inscript spheres

V.c 43 / π * 32 / sqrt R * (3^* : 492.4991

V.i 43 / π * R 3^* : 268.0826

Alvaro

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