Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
**Figure 2.** A torus.
This image shows a typical torus, a donut-shaped object, where \( R \) is the distance from the center of the tube to the center of the torus, and \( r \) is the radius of the tube.
### Task a:
**Use the method of shells to find the volume of this torus.**
#### Figure 3:
**Figure 3.** Using the method of shells.
This diagram illustrates the method of cylindrical shells applied to the given problem. The circle \((x - R)^2 + y^2 = r^2\) is revolved about the y-axis to form the torus. Points of interest include:
- \((R-r,0)\): the leftmost point of the circle.
- \((R+r,0)\): the rightmost point of the circle.
- The cylindrical shell element is shown in red, extending a distance \(x\).
### Task b:
**Use the Theorem of Pappus to find the volume of this torus.**
The **Theorem of Pappus** states that the volume \( V \) of a solid of revolution is the product of the area \( A \) of the shape being revolved and the distance \( d \) traveled by the centroid of this shape during the revolution.
### Graph and Diagram Explanation:
1. **Figure 2 (Torus)**:
- A 3D image depicting a torus.
- Represents \( R \) as the major radius (from the center of the tube to the center of the torus).
- Represents \( r \) as the minor radius (radius of the tube).
2. **Figure 3 (Method of Shells)**:
- A 2D graph to apply the method of cylindrical shells.
- Shows the circle described by \((x - R)^2 + y^2 = r^2\).
- The red cylindrical shell segment visualizes an infinitesimally thin slice of the torus.
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