Problem 215 - Calculus Volume 1

Question

A regular \(N\) -gon (an \(N\) -sided polygon with sides that have equal length \(s,\) such as a pentagon or hexagon) has perimeter \(N s .\) Write an integral that expresses the increase in perimeter of a regular \(N\) -gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

Step-by-Step Solution

Answer

The increase in perimeter is \(N\) units.

Step 1: Define the Function

First, note that the function for the perimeter of the regular \(N\)-gon is \(P(s) = N \times s\), where \(s\) is the length of each side and \(N\) is the number of sides.

Step 2: Set Up the Integral

To find the increase in perimeter as the side length \(s\) changes from 1 to 2 units, set up the integral: \[ \int_{1}^{2} N \, ds. \] This integral represents the total change in the perimeter over the specified interval of \(s\).

Step 3: Evaluate the Integral

Evaluate the integral \(\int_{1}^{2} N \, ds\). The integral of a constant \(N\) with respect to \(s\) is \(N \times s\): \[ \int_{1}^{2} N \, ds = \left[ N \times s \right]_{1}^{2}. \]

Step 4: Calculate the Definite Integral

Apply the limits of integration to find the result: \[ \left[ N \times s \right]_{1}^{2} = N \times 2 - N \times 1 = 2N - N = N. \] This means the increase in perimeter is \(N\) units.

Key Concepts

Definite Integral

A definite integral is like a mathematical lens focusing on the area under the curve of a function for a specified interval. In plain terms, it helps us measure how much change happens as we move from one point to another. In the context of our exercise, the definite integral is used to compute the increase in perimeter as the side length changes. The integral

• \( \int_{1}^{2} N \, ds \)

helps us find the total change in the perimeter when each side length of a regular \(N\)-gon expands from 1 to 2 units.
In this specific problem, the integral gives us the exact difference, or increase, in the perimeter due to the change in side length. This kind of precise calculation is one of the many powerful applications of definite integrals in calculus.

Perimeter

Think of the perimeter of a polygon as the total distance around its edges. For a regular polygon, such as a pentagon or hexagon, known as an \(N\)-gon, each side has the same length. Hence, a quick way to find the perimeter is by multiplying the number of sides \(N\) by the side length \(s\). The formula used is

• \( P(s) = N \times s \)

In our problem, initially, each side was 1 unit long. When these lengths increase to 2 units, we calculate the expansion in the perimeter by assessing the difference between the perimeters at these two states using integration. This showcases how a perimeter function can be used dynamically, based on changing conditions.

Polygons

A polygon is a flat shape consisting of straight, non-intersecting lines that make a closed chain. Some common polygons include triangles, squares, and hexagons. A regular \(N\)-gon is a special polygon where all sides and angles are equal. This regularity simplifies calculations such as perimeter and area, as we see with our perimeter formula.
Understanding polygons is crucial because they form the basis of many geometric figures, which appear in various mathematical and real-world applications. The uniform nature of regular polygons means that integrating or finding exact changes in their properties (like perimeter) is more straightforward compared to irregular polygons.

Calculus

Calculus is a field of mathematics that explores how things change. It provides tools and techniques like derivatives and integrals that help us study the behavior of functions. One major aspect of calculus is its ability to solve problems related to rates of change and accumulation, like the increase in perimeter of a polygon.
In this exercise, calculus, through definite integration, helps quantify the perimeter's change over a certain interval. By integrating the constant function representing the perimeter with respect to \(s\), we find precise changes in the geometric properties of shapes like polygons. This integration not only helps in theoretical problems but also in practical situations where changes are continuous and require a precise measure.