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

48879e23-eff7-4188-af2d-b776ccd97aa3 65 overlord 4 5 false false false 0 3 decimal

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Steinmetz Solid Volume

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 :

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.4 4 :

R.1 1 :

k R V θ φ f k r (3^3 / θ sin * abs eval : 8 θ φ f 0 π 2 / 0 π 2 / 2000020000 DInt * 21 line :

Volume

Vx err V Vx - abs Vx / :

k.2 1212 mat :

V k.2 R.4 V : 341.3544

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

V.2 err % 0.01

k.3 12313 mat :

V k.3 R.4 V : 300.0283

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

V.3 err % 0.04

k.4 456714 mat :

V k.4 R.1 V : 4.5466

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

V.4 err % 0.01

k.6 813 range :

V k.6 R.1 V : 4.3041

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

V.6 err % 0.03

k.7 17 range :

V k.7 R.1 V : 4.2803

Maple code

restart: with(linalg):
c := 1 / sqrt(3): d := 1 / sqrt(2):
U := matrix([
[1, 0, 0, c, c, c, -c, d, d, 0, 0, d, d],
[0, 1, 0, c, c, -c, c, d, -d, d, d, 0, 0],
[0, 0, 1, c, -c, c, c, 0, 0, d, -d, d, -d]
]):
n := vector([sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)]):
V := proc(K, R)
local N, r, f;
N := [seq(norm(crossprod(n, col(U, idx)), 2), idx in K)];
r := R / max(op(N));
f := abs(r^3 * sin(theta) / 3);
return 8 * evalf(Int(Int(f, theta = 0 .. Pi/2), phi = 0 .. Pi/2, epsilon = 1e-3));
end proc:

R_4 := 4: R_1 := 1:

V2 := V([1, 2], R_4);
V3 := V([1, 2, 3], R_4);
V4 := V([4, 5, 6, 7], R_1);
V6 := V([8, 9, 10, 11, 12, 13], R_1);
V7 := V([1, 2, 3, 4, 5, 6, 7], R_1);

Alvaro

4 appVersion 1.75.9678.0