<content> <p>DInt</p> </content>

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Steinmetz Solid

Refrerences:

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

https://mathworld.wolfram.com/SteinmetzSolid.html

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_ExplicitRK45 : 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 :

Unit vectors of cylinders

c 13 sqrt / :

d 12 sqrt / :

U 100 c c c c - d d 00 d d 010 c c c - c d d - d d 00001 c c - c c 00 d d - d d - 313 mat :

k u U k col :

Normal in spherics & Position vector of the surface

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

k r R N k el n k u † norme : max / :

Radius of the cylinders

R.1 1 :

R.4 4 :

CreateMesh mathcad :

Surface plot and Volume value

k R M θ φ S k r 3 identity * n * : θ φ S 02 π * 0 π 5050 CreateMesh 21 line :

k R V θ φ f k r (3^3 / θ sin * abs eval : θ φ f 02 π * 0 π 10001000 DInt 21 line :

k.2 1212 mat :

k.3 12313 mat :

k.2 R.4 M 1 1 line k.3 R.4 M 1 1 line

k.2 R.4 V 342.0346

k.3 R.4 V 300.1985

V.2 163 / R.4 3^* : 341.3333

V.3 822 sqrt - (* R.4 3^* : 299.9227

k.4 456714 mat :

k.6 891011121316 mat :

k.4 R.1 M 1 1 line k.6 R.1 M 1 1 line

k.4 R.1 V 4.5493

k.6 R.1 V 4.304

V.4 1222 sqrt * 6 sqrt - (* : 4.5472

V.6 163 / 323 sqrt * 42 sqrt * - + (* : 4.3053

Alvaro

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