Introduction to Trapezoidal Rule in Engineering

The trapezoidal rule is a fundamental numerical integration technique that is widely used in engineering and science. It is particularly useful for approximating the definite integral of a function. One practical application involves calculating the area of irregular shapes, such as the area of a door rounded at the top. In this article, we will explore how to apply the trapezoidal rule to solve an authentic engineering problem related to the weight and paint requirements for such a door.

Data and Problem Statement

Consider a home door that is rounded at the top, as illustrated in the accompanying diagram. To estimate its weight and paint requirements, we need to determine the area of this rounded door. If we divide the door into narrow horizontal strips, we can approximate the area of each strip using the trapezoidal rule. This method provides an accurate estimation with minimal error, making it a valuable tool in engineering design and analysis.

Step-by-Step Guide to Applying the Trapezoidal Rule

The trapezoidal rule involves dividing the area under a curve into trapezoids. The formula for the trapezoidal rule is given by:

Area ≈ (h/2) [f(x0) 2f(x1) 2f(x2) ... 2f(x(n-1)) f(xn)]

where h is the width of each strip, and f(x) represents the height of the door at each strip.

Example Problem: Calculating the Area of a Rounded Door

Step 1: Define the Problem

Imagine a door with a rounded top. Let's assume the height of the door is 2 meters, and the top is rounded with a radius of 0.3 meters. We will divide the door into 6 horizontal strips of equal width.

Step 2: Gather Data

The height of each strip at the top curved part will vary linearly with the width of the door. We can use the following data to calculate the area:

Step 3: Apply the Trapezoidal Rule

Using the trapezoidal rule formula, we get:

Area ≈ (0.2/2) [1.2 2(1.1) 2(1.0) 2(0.9) 2(0.8) 0.8]

Area ≈ 0.1 [1.2 2.2 2 1.8 1.6 0.8]

Area ≈ 0.1 [9.6]

Area ≈ 0.96 square meters

Step 4: Interpret the Results

The total area of the rounded door is approximately 0.96 square meters. Using this area, we can now estimate the weight and paint requirements.

Weight of the Door

The weight of the door depends on its density. Let's assume the density of the wood is 600 kg/m3. The weight can be calculated as follows:

Weight Area × Density 0.96 m2 × 600 kg/m3 576 kg

Paint Requirements

To estimate the paint requirements, we need to determine the surface area of the door. Assuming the door is 1 meter wide, the total surface area is:

Total Surface Area Height × 2 Width × 2 (for the sides) 2 meters × 2 1 meter × 2 6 square meters

Petroleum-based paint typically requires 10-15 square meters per liter. For 6 square meters, we would require 0.4 to 0.6 liters of paint.

Comparison with Other Integration Techniques

The trapezoidal rule provides a good approximation for this problem, but other techniques like Simpson's rule or Boole's rule can sometimes offer more accurate results depending on the complexity of the curve.

Simpson's Rule

Simpson's rule is more accurate than the trapezoidal rule but requires more data points. For a quadratic function, Simpson's rule can achieve zero error. The formula for Simpson's rule is:

Area ≈ (h/3) [f(x0) 4f(x1) 2f(x2) 4f(x3) 2f(x4) ... 4f(x(n-1)) f(xn)]

Boole's Rule

Boole's rule is an even more accurate method, particularly for highly curved functions. The formula for Boole's rule is:

Area ≈ (73/90)h [f(x0) 32f(x1) 12f(x2) 32f(x3) 7f(x4)]

While these methods provide higher accuracy, they also require more computational effort and data points.

Conclusion

The trapezoidal rule is a valuable tool in engineering applications, especially for approximating the area of irregular shapes. By dividing the door into narrow horizontal strips and applying the trapezoidal rule, we can accurately estimate the area, weight, and paint requirements. While other methods like Simpson's rule and Boole's rule offer higher accuracy, the trapezoidal rule remains a practical and efficient choice for many engineering problems.

Keywords: trapezoidal rule, engineering problems, Simpsons rule, Boole’s rule, area calculation