RQ: How can integration be used to determine the surface area of a soda bottle

Introduction

As a student passionate about protecting the environment, I have always been curious about how much plastic is used in the production of various bottles and the implications of their surface areas. Research indicates that plastic waste can take anywhere from 20 to 500 years to decompose, and even then, it does not fully disappear, posing a long-term threat to ecosystems. Recognizing the environmental impact of plastic, I chose to investigate the surface area of a 2-litre soda bottle. By applying calculus concepts learned both in and outside the classroom, this exploration aims to calculate the bottle’s surface area and emphasize the broader implications of plastic usage in our daily lives.

Aim

The aim of this experiment is to investigate using calculus about the surface area of a 2-litre soda bottle.

Data Collection

The Soda Bottle (2 Liter) has an overall height of 12.4” (31.5 cm) and a diameter of 4.33” (11 cm).

Measurement

Calculation

Separate the bottle into three different parts to calculate the surface area

This exploration successfully applied integration to calculate the surface area of a 2-litre soda bottle, with the focus primarily on the curved body of the bottle. The calculation revealed that calculus offers an effective tool for understanding and quantifying physical properties of real-world objects. This investigation also emphasized the importance of precise mathematical modeling and accurate data collection to achieve reliable results.

From an environmental perspective, the study highlights the implications of plastic usage by providing a measurable metric—the surface area—to estimate the volume of plastic material used in bottle production. This encourages further consideration of sustainable alternatives and waste reduction strategies.

Part 1 calculation:

To calculate the area of Part 1, we have to derive the surface area formula that is obtained by revolution (i.e. area of surface of revolution).

Mathematically, let f(x) be a differentiable and non-negative function on the interval [a, b], we wish to find the surface area by revolving the curve of y = f(x) from x = a to x= b around the x-axis (see the following diagram for reference).

Using a similar strategy we learned when trying to find the area under the curve, we are going to partition the interval [a, b] into n equal sub-intervals. On every ith sub-interval (where l ≤ i ≤ n), draw a line segment from (xi-1, f(xi-1)) to (xi, f(xi)) then revolve all these line segments around the x-axis we will get the following:

Notice that after revolving each line segment, we get a trapezoidal-like shape, or a band (illustrated as the shaded region in the diagram above on the right). The formal name of this geometric shape is called a frustum of a cone.

Before finding the total surface area, we start by first finding the area of one frustum:

Let r1 and r2 be the radii of the bottom part and upper part of the frustum, respectively. Let l be the slant height of the frustum, as shown in the following diagram.

A frustum can be thought of as a complete cone but with the upper sharp part removed. Thus, we are going to calculate the total surface area of a cone minus the surface area of the top part, as shown in the following diagram.

Next, we wish to express s in terms of the remaining variables r1, r2 and l. By using similar triangles, we have:

Now we can substitute this expression of s to the previous equation, and we have the following:

This is a huge breakthrough and we are almost there! Now back to the context of interval partition. in the formula corresponds to length of each line segment we connected from (xi-1, f(xi-1)) to (xi, f(xi), r1 and r2 represent the height of the curve at (xi-1, f(xi-1)) and (xi, f(xi)), respectively.

Therefore, the “refined” surface area formula is:

Since we assume our function f(x) is differentiable on [a, b], it is also differentiable on every sub-interval [xi-1, xi]. By the Mean Value Theorem, there must exist an xi* on (xi-1, xi) such that

With this, we can further simplify the above formula to:

Furthermore, using the Intermediate Value Theorem, there must also exist another point, say xi** ∈ [xi-1, xi] such that f(xi**) = 2/1[f(xi-1 + f(xi)]. So the above formula can be simplified again as:

Hence, the total surface area can be approximated by summing over all the frustum.

Mathematically

Thus, the exact area can be obtained by taking the integral:

And this is the formula for calculating surface area of revolution of a curve f(x) over the interval [a, b].

For the curve y= f1(x)

To interpret this graph from my curve equation:

The quadratic equation that passes through the points E=(7.3468347739241,14.784138488014)

K=(19.8332914008764,9.1211798726049)

C = (0, 17.4293972353711)

y =f(x)=  -0.0047x2 - 0.3254x + 17.4294

R=(-1.9372858830404,17.3491541265612)

O=(28.6197020391998,4.0410369944474)

x(R)≤ x ≤ x(0)

Evaluation

The application of integration to determine the surface area of a soda bottle demonstrated the practical use of calculus in analyzing real-world objects. The approach of dividing the bottle into three parts—the main cylindrical body, the neck, and the base—enabled the calculations to be more manageable and accurate. By using a derived quadratic equation to approximate the curve of the bottle, the calculations for Part 1 revealed a surface area of 2483.10 square centimeters.

However, several factors influenced the accuracy of these results:

Curve Approximation: The quadratic equation used to model the curve of the bottle is an approximation, which might not perfectly represent the actual shape. Small deviations in the shape can lead to discrepancies in the calculated surface area.

Measurements: The measurements of the soda bottle were taken manually, introducing potential human error. More precise tools, such as calipers or laser measurements, could improve accuracy.

Simplifications: The calculations assumed smooth transitions between different sections of the bottle. In reality, sharp edges or indentations, particularly at the base and neck, were ignored.

Integration Assumptions: The formula used for surface area relies on accurate evaluation of the integral. Computational errors in handling the quadratic equation or its derivative could affect the final result.

Despite these limitations, the methodology provided a reasonable estimate of the bottle’s surface area and highlighted the significant role of calculus in environmental studies.

Conclusion

This exploration successfully applied integration to calculate the surface area of a 2-litre soda bottle, with the focus primarily on the curved body of the bottle. The calculation revealed that calculus offers an effective tool for understanding and quantifying physical properties of real-world objects. This investigation also emphasized the importance of precise mathematical modeling and accurate data collection to achieve reliable results.

From an environmental perspective, the study highlights the implications of plastic usage by providing a measurable metric—the surface area—to estimate the volume of plastic material used in bottle production. This encourages further consideration of sustainable alternatives and waste reduction strategies.

Limitations

Curve Modeling Accuracy: The quadratic function used to represent the bottle’s shape does not account for small imperfections or irregularities in the actual design.

Precision of Measurements: Manual measurements of the bottle dimensions could introduce errors.

Neglected Sections: Certain features, such as the cap and fine details of the neck or base, were not included in the calculation.

Numerical Integration: Rounding errors and approximations in the integration process may affect the final values.

Material Thickness: The calculation assumes the bottle is a hollow object, ignoring variations in material thickness.

Areas of Further Research

Incorporating 3D Scanning Technology: Using 3D scanning to capture the exact dimensions and curves of the bottle would improve the accuracy of the model.

Material Usage Analysis: Extending the study to investigate the thickness and type of plastic used, providing a more comprehensive understanding of resource usage.

Comparison Across Bottles: Analyzing the surface areas of different bottle designs to evaluate their environmental efficiency.

Sustainability Impact: Investigating alternative materials and their respective surface areas to assess environmental benefits.

Volumetric Analysis: Exploring how the surface area correlates with the bottle’s internal volume to evaluate material efficiency.