Center of Mass

Begin with the equation of a circular cylinder:
- x^2 + y^2 = 4
- The cylinder has a radius of 2.
- Domain of x: -2 \leq x \leq 2
- y values lead to both positive and negative square roots of y.
- Area of integration is within the cylinder, represented in the xy-plane.
Note: The footprint (area) defined by x^2 + y^2 = 4 is not part of the solid over which we are integrating.

Visualizing the cylinder:
- It encompasses all points inside a circle of radius 2 on the xy-plane.
Analysis of z values:
- Rearranging gives:
- z = 2 \pm \sqrt{4 - x^2 - y^2}
- The equation represents a sphere centered at another point. Specifically, it translates to:
- Relation: x^2 + y^2 + (z - 2)^2 = 4
- This shape resembles a ball positioned above the origin, resting on the xy-plane.

Spherical Coordinates:
- Transitioning into spherical coordinates involves using:
- x = \rho \sin \phi \cos \theta
- y = \rho \sin \phi \sin \theta
- z = \rho \cos \phi
Upon squaring the coordinates, the simplification leads to:
- \rho^2 - 4z = 0
Transforming to spherical coordinates gives us the equation:
- \rho = 4 \cos \phi
- Which describes a sphere on the xy-plane.

When analyzing the angle phi:
- At \phi = 0, \rho = 4 which is at the top of the sphere.
- As \phi increases, \rho shrinks, demonstrating the volume of integration.
Limitations:
- Avoid values where \rho becomes negative; valid limits only exist if the sphere remains above the xy-plane.

Importance of maintaining the volume element for integration:
- dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta
Function for integration, since x^2 + y^2 + z^2 translates into spherical coordinates as:
- \rho^2
The integral involves calculating:
- \int \rho^3 \, dV
- Result: \frac{4096}{21} \pi (simplified result of integration).

Starting point: Integrals can be applied to find mass when combined with density functions.
Density Function (Coefficient of mass density):
- Notation: \delta for two-dimensional analysis.
- Indicates mass density in area (grams per square centimeter).
The double integral of a density function gives the total mass:
- For mass calculation, consider tiny rectangle areas, mass \approx density \times area.

First Mass Moment:
- Notated as M_y, represents value weighted by distance from y-axis.
Moment statistical description:
- Involves density and distance tallies.
- Approach: Integrate the density function weighted by the respective coordinate (x or y).
To find the center of mass:
- Use the relations \bar{x} = \frac{My}{m} and \bar{y} = \frac{Mx}{m} where m is the total mass.

Consider the curve y = x and the area between 0\leq x \leq 1.
Density function is proportional to distance from the y-axis (thus kx).
Integrals needed to find mass and moments:
- m = k \int_0^1 (x^2)dx.
- Final calculation indicates \frac{k}{12} for mass.
Calculation involves repeated integration for moments, leading to coordinates of the center of mass:
- Result: Center of mass located at \left(\frac{3}{5}, \frac{1}{2} \right)

Explore triple integrals focusing on volumes, specifically tetrahedrons:
- Example defined by density function over a triangular base.
Integrations must be set up correctly to account for volume:
- mass = \int \int \int \delta(x,y,z) \, dz \, dy \, dx
- For valid regions, arrange appropriate limits based on the geometric shape.
Employ density functions to approximate mass accurately - yields real-life applications in physics and engineering contexts.

Review forming integrals in various coordinate systems, recognizing changes in geometry.
Maintain clarity in definitions and transformations for smooth calculations in multiple dimensions.