3D Volume Visualization and Integration in Calculus

Graphical Representation of Volume in the First Octant

The following commands generate a graphical representation of a volume defined in the first octant (where $x, y, z \ge 0$).

Defining 3D Surfaces

• Cylinder 1 (Cil_1): implicitplot3d (x ^ 2 + y ^ 2 - 2 * x = 0, x = 0 .. 2, y = 0 .. 2, z = 0 .. 4, color = GREEN);
• Cylinder 2 (Cil_2): implicitplot3d (x ^ 2 + y ^ 2 - 4 * x = 0, x = 0 .. 4, y = 0 .. 4, z = 0 .. 4, color = BLUE);
• Cone: implicitplot3d (x ^ 2 + y ^ 2 - z ^ 2 = 0, x = 0 .. 2, y = -2 .. 2, z = 0 .. 4, color = RED);

Displaying the Plot

Display the defined objects:

Display ([cil_1, cil_2, cone]);

Substitution Check: We verify whether a bridge (implied boundary) falls below the cone. In this case, it falls below the cone.

Cylindrical Coordinate Transformation for Mass Calculation

Defining the Density Function

First, defining the initial density function:

F: = (x, y, z) -> y;

Transforming the First Cylinder Equation

We transform the equation for the first cylinder using cylindrical coordinates ($x = r \cos(t), y = r \sin(t)$):

1. Subs (x = r * cos (t), y = r * sin (t), x ^ 2 + y ^ 2 - 2 * x = 0);
2. Simplify (%);
3. Solve ((- (2 * cos (t) - r) * r = 0), (r));

This yields the lower limit of the radius: $r = 2 \cos(t)$.

Transforming the Second Cylinder Equation

Transforming the equation for the second cylinder:

1. Subs (x = r * cos (t), y = r * sin (t), x ^ 2 + y ^ 2 - 4 * x = 0);
2. Simplify (%);
3. Solve ((- (4 * cos (t) - r) * r = 0), (r));

Transforming the Cone Equation

Transforming the cone equation:

1. Subs (x = r * cos (t), y = r * sin (t), x ^ 2 + y ^ 2 - z ^ 2 = 0);
2. Simplify (%);
3. Solve ((-z ^ 2 + r ^ 2 = 0), (z));

Defining the Mass Integral (MASA)

Now, define the integral for mass calculation using the derived limits:

Int (Int (Int (r * f (r * cos (t), r * sin (t)), z = 0 .. r), r = 2 * cos (t) .. 4 * cos (t) ), t = 0 .. Pi / 2);

A second attempt using lowercase functions:

Int (int (int (r * f (r * cos (t), r * sin (t)), z = 0 .. r), r = 2 * cos (t) .. 4 * cos (t) ), t = 0 .. Pi / 2);

Alternative Density Function and Spherical Coordinates

The density function is defined as the distance from a point to the plane (perpendicular to the plane projection): $\rho = Kz$, where $K$ is any real number.

We switch to spherical coordinates as Cartesian coordinates yield many roots.

Defining Density and Spherical Transformations

• Density: (x, y, z) -> k * z;
• Spherical Variables: (Defined as variables, not functions)
  • X: = r * cos (f) * cos (t);
  • Y: = r * cos (f) * sin (t);
  • Z: = r * sin (f);
• Jacobian: Jacob: = r ^ 2 * cos (f);

Mass Integral in Spherical Coordinates

Setting up the integral for mass:

Int (Int (Int (density (x, y, z) * jacob, t = 0 .. 2 * Pi), f = 0 .. Pi / 2), r = 0 .. 2) = int (int (int (density (x, y, z) * jacob, t = 0 .. 2 * Pi), f = 0 .. Pi / 2), r = 0 .. 2);

Moments of Inertia

Axis OX

Calculating the moment of inertia about the OX axis:

Int (Int (Int ((y ^ 2 + z ^ 2) * density (x, y, z) * jacob, r = 0 .. 2), f = 0 .. Pi / 2), t = 0 .. 2 * Pi) = int (int (int ((y ^ 2 + z ^ 2) * density (x, y, z) * jacob, r = 0 .. 2), f = 0 .. Pi / 2) , t = 0 .. 2 * Pi);

Axis OY

Calculating the moment of inertia about the OY axis:

Int (Int (Int ((x ^ 2 + z ^ 2) * density (x, y, z) * jacob, r = 0 .. 2), f = 0 .. Pi / 2), t = 0 .. 2 * Pi) = int (int (int ((x ^ 2 + z ^ 2) * density (x, y, z) * jacob, r = 0 .. 2), f = 0 .. Pi / 2) , t = 0 .. 2 * Pi);