O17-C031
> restart:
> f:=22*x^2+22*y^2+19*z^2+8*x*y-4*x*z-4*y*z-148*x-4*y+236*z-1252;

           2       2       2
  f := 22 x  + 22 y  + 19 z  + 8 x y - 4 x z - 4 y z - 148 x - 4 y

         + 236 z - 1252

> gra:={diff(f,x),diff(f,y),diff(f,z)};

  gra := {44 x + 8 y - 4 z - 148, 44 y + 8 x - 4 z - 4,

        38 z - 4 x - 4 y + 236}

> solve(gra);

                       {x = 3, y = -1, z = -6}

> f1:=expand(subs(x=x1+3,y=y1-1,z=z1-6,f));

                     2        2        2
  f1 := -2180 + 22 x1  + 22 y1  + 19 z1  + 8 x1 y1 - 4 x1 z1

         - 4 y1 z1

> with(LinearAlgebra);

  [&x, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis,

        BezoutMatrix, BidiagonalForm, BilinearForm,

        CharacteristicMatrix, CharacteristicPolynomial, Column,

        ColumnDimension, ColumnOperation, ColumnSpace,

        CompanionMatrix, ConditionNumber, ConstantMatrix,

        ConstantVector, Copy, CreatePermutation, CrossProduct,

        DeleteColumn, DeleteRow, Determinant, Diagonal,

        DiagonalMatrix, Dimension, Dimensions, DotProduct,

        EigenConditionNumbers, Eigenvalues, Eigenvectors, Equal,

        ForwardSubstitute, FrobeniusForm, GaussianElimination,

        GenerateEquations, GenerateMatrix, Generic, GetResultDataType,

        GetResultShape, GivensRotationMatrix, GramSchmidt,

        HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm,

        HilbertMatrix, HouseholderMatrix, IdentityMatrix,

        IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar,

        IsUnitary, JordanBlockMatrix, JordanForm, KroneckerProduct,

        LA_Main, LUDecomposition, LeastSquares, LinearSolve, Map,

        Map2, MatrixAdd, MatrixExponential, MatrixFunction,

        MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower,

        MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial,

        Minor, Modular, Multiply, NoUserValue, Norm, Normalize,

        NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm,

        QRDecomposition, RandomMatrix, RandomVector, Rank,

        RationalCanonicalForm, ReducedRowEchelonForm, Row,

        RowDimension, RowOperation, RowSpace, ScalarMatrix,

        ScalarMultiply, ScalarVector, SchurForm, SingularValues,

        SmithForm, StronglyConnectedBlocks, SubMatrix, SubVector,

        SumBasis, SylvesterMatrix, ToeplitzMatrix, Trace, Transpose,

        TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd,

        VectorAngle, VectorMatrixMultiply, VectorNorm,

        VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip]

> a:=Matrix(3,3,[22,4,-2,4,22,-2,-2,-2,19]);

                             [22     4    -2]
                             [              ]
                        a := [ 4    22    -2]
                             [              ]
                             [-2    -2    19]

> Eigenvalues(a);

                                 [27]
                                 [  ]
                                 [18]
                                 [  ]
                                 [18]

27 x2^2+18 y2^2+18 z2^2=2180
> a:=sqrt(27/2180);b:=sqrt(18/2180);r:=b*sqrt(1-z^2/27*2180);

                                       1/2
                                 3 1635
                            a := ---------
                                   1090


                                       1/2
                                 3 1090
                            b := ---------
                                   1090


                             1/2             2 1/2
                         1090    (81 - 6540 z )
                    r := -------------------------
                                   3270

> with(plots);

  [animate, animate3d, animatecurve, arrow, changecoords, complexplot,

        complexplot3d, conformal, conformal3d, contourplot,

        contourplot3d, coordplot, coordplot3d, densityplot, display,

        dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d,

        graphplot3d, implicitplot, implicitplot3d, inequal,

        interactive, interactiveparams, intersectplot, listcontplot,

        listcontplot3d, listdensityplot, listplot, listplot3d,

        loglogplot, logplot, matrixplot, multiple, odeplot, pareto,

        plotcompare, pointplot, pointplot3d, polarplot, polygonplot,

        polygonplot3d, polyhedra_supported, polyhedraplot, rootlocus,

        semilogplot, setcolors, setoptions, setoptions3d, spacecurve,

        sparsematrixplot, surfdata, textplot, textplot3d, tubeplot]

> plot3d(r, theta=-Pi..Pi, z=-a..a, coords=cylindrical);

> aire:=int(2*Pi*r*sqrt(1+(diff(r,z))^2),z=-a..a);

                                                   1/2
                       9 Pi    27      1/2        3
               aire := ---- + ---- Pi 2    arcsin(----)
                       545    1090                 3

> evalf(aire);evalf(a);evalf(b);

                             0.1196149393


                             0.1112893575


                            0.09086737993

> evalf(4*Pi*0.01);

                             0.1256637062

>