4. The fractal called the snowflake island (or Koch island) is constructed as follows: Let Io be an equilateral triangle with sides of length 1. The figure I1 is obtained by replacing the middle third of each side of Io with a new outward equilateral triangle with sides of length 1/3 (see figure). The process is repeated, where In+1 is obtained by replacing the middle third of each side of I, with a new outward equilateral triangle 1 The limiting figure as n→o is called the snowflake island. with sides of length 3n+1* (a) Let L, be the perimeter of In. Show that lim L, = ∞ (b) Let A, be the area of Im. Find lim An. (It exists!)
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.
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