Problem 213 - Calculus Volume 1

Question

Write an integral that expresses the increase in the perimeter \(P(s)\) of a square when its side length \(s\) increases from 2 units to 4 units and evaluate the integral.

Step-by-Step Solution

Answer

The increase in the perimeter is 8 units.

Step 1: Understand the Problem

To find the increase in the perimeter of a square as its side length changes from 2 to 4 units, we need to express the perimeter as a function of the side length and identify the integral that represents this change.

Step 2: Define the Perimeter Function

The perimeter \( P(s) \) of a square with side length \( s \) is defined by the equation \( P(s) = 4s \), because a square has four equal sides.

Step 3: Set Up the Integral Expression

To find the increase in the perimeter as \( s \) increases from 2 to 4, we set up the integral \[ \int_{2}^{4} P'(s) \, ds \] where \( P'(s) = \frac{d}{ds}(4s) = 4 \) represents the constant rate of change of the perimeter with respect to \( s \).

Step 4: Evaluate the Integral

Evaluate \[ \int_{2}^{4} 4 \, ds \]. This simplifies to \[ 4[s]_{2}^{4} = 4(4 - 2) = 4 \times 2 = 8 \].

Step 5: Interpret the Result

The evaluation of the integral gives us the increase in the perimeter, which is 8 units. This means that as the side length of the square increases from 2 units to 4 units, the perimeter increases by 8 units.

Key Concepts

Perimeter of a Square

A square is a simple geometric shape, and its perimeter is the total distance around its edges. Calculating the perimeter is straightforward because a square has four equal sides. The formula for the perimeter of a square is:

• \( P(s) = 4s \)

Here, \( s \) represents the length of one side of the square. This formula is crucial because it shows that the perimeter is directly proportional to the side length. When you know the length of one side, you just multiply it by 4 to find the perimeter. Understanding this concept helps you express the perimeter as a function of the side length, which is essential in calculus for analyzing changes in geometric shapes. It's important to note how small changes in the side can significantly affect the perimeter of larger squares.

Rate of Change

In mathematics, the rate of change refers to how a quantity changes over a specific interval. For a square, if the side length changes, the perimeter changes at a predictable rate. Specifically, when you differentiate the perimeter \( P(s) = 4s \) with respect to \( s \), you find that the rate of change of the perimeter is a constant.

• \( P'(s) = \frac{d}{ds}(4s) = 4 \)

This tells us that for every unit increase in side length, the perimeter increases by 4 units. This constant rate of change simplifies the process of understanding how alterations in side lengths affect the perimeter. It eliminates the need for complex calculations and sets the stage for analyzing this rate over different intervals using integrals.

Evaluating Integrals

Evaluating integrals is a fundamental tool in calculus used to calculate accumulated quantities, such as area under a curve or the total change between two points. An integral, like \( \int_{a}^{b} f(x) \, dx \), represents the sum of infinitesimal changes from \( a \) to \( b \). In our example with the square's perimeter, the integral \( \int_{2}^{4} 4 \, ds \) computes the total increase in the perimeter as the side length increases from 2 to 4.
The function \( 4 \) is constant, simplifying the integral to:

• \( 4[s]_{2}^{4} = 4(4 - 2) = 8 \)

This result indicates that over the interval from 2 to 4 units, the perimeter rises by 8 units. Grasping integral evaluation allows us to measure changes and the cumulative effect of these changes across an interval, which is particularly useful in physics, engineering, and various fields requiring precise measurements of dynamic systems.