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β€œI want the project to be related to how calculus is used in mechanical engineering” What the project should include: Your project should feature a practical problem from the field you are pursuing and require the use of calculus tools. Beyond that, problems might be either well-defined or Open-ended. Your project (equations, graphs, diagrams, pictures included) should be presented as a Microsoft WORD document. Clarity of writing is important. At the very least, be sure to use your spell-checking and grammar-checking facilities. Report Format (a) Cover page and Problem statement The cover page should use the following template, followed by the problem statement. PROBLEM STATEMENT Provide an exact statement of the problem as suggested by its author. (b) Table of Contents Include the following sections in the table and give the page numbers. Contents: 1. Abstract 2. Motivation 3. Mathematical Description and Solution Approach 4. Discussion 5. Conclusions and Recommendations 6. Nomenclature 7. References Appendix (calculations, graphs, pictures, spreadsheet information ) (c) Abstract The abstract is a short summary of your project report; it should not exceed one or two paragraphs. It should concisely state what you did, how you did it, and what conclusions you drew from the results. (d) Motivation In this section you should give some background about why the problem is important to science or engineering. You should also describe the problem within its engineering or science context and provide the objective for the project. (e) Mathematical Description and Solution Approach In this section, you should formulate the mathematical approach to solving the problem providing the relevant equations, describing the mathematical tools you used and outline the procedure used. Do NOT simply list the equations use text between them to provide a clear understanding of them to the reader. (f) Discussion Here, you should provide the results and discuss them. Did you meet the objective of the project? Were they as expected, or were they counter-intuitive? What implications do your results have to the problem at hand and to the field in general? (g) Conclusions and Recommendations Give the basic conclusions of your work. This will be somewhat similar to what is in the abstract but with a little more detail for instance, including a summary of your interpretation of the results. You should also make a few recommendations such as things a person doing the same project might do differently or ideas for a new study that is suggested by your results. (h) Nomenclature List the symbols that you use in your report. For each symbol, provide a description of what it represents and its units. All units used should belong to the same measuring system: Standard (English) or Metric (SI). Carefully check whether the units agree and are balanced on both sides of each equation. (i) References Any work or ideas that you have taken from someone else should be cited directly in the text of your report. This includes any figures that you might download from the web. Do your best to find and cite the original source of information rather than the secondhand source. At the end of the report should be a list of references that were cited. Book and scientific journal references are strongly preferable to webpages. (j) Appendices You might have detailed calculations, spreadsheets or computer programs that were used to obtain your results but do not belong in the main report. If so, you should place these materials in appendices and refer to them as needed in the report. Area of Irregular Shaped Roof Abstract This project makes use of calculus, specifically integration, to find the exact area of a roof garden to minimize the cost of material used in flooring it. Since the roof is of irregular shape, it is divided into four sections, the integral of each section is taken, and a relationship between the integrals is generated. Positive results are found, and the data reveal that the area of the grass floor is 175.3 m2 while the area of the wooden floor is 23.4 m2 with the total price being 2472 US dollars. Keywords irregular shape, area, integration, cost of material Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. This article is available in Undergraduate Journal of Mathematical Modeling: One + Two: Alkhawaldeh: Area of Irregular Shaped Roof 1 PROBLEM STATEMENT Find the exact area of the rooftop demonstrated by the floor plan below (Figure 1) for the purpose of determining the accurate amounts and costs of wood and artificial grass needed to floor it. Figure 1: The floor plan of the roof top to be turned into a garden Produced by The Berkeley Electronic Press, 2021 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 11, Iss. 2 [2021], Art. 3 2 MOTIVATION With all the trouble that COVID-19 has brought, going out to restaurants and parks became less wanted by people. Staying at home and quarantining became the solution. While staying at home, people tend to explore their houses and make them a comfortable place to stay. After noticing the empty spaces in their house, many people decide to contact house architects to help them out. One example of a famous design is a roof garden, where kids and pets can play without the fear of contacting unknown people (see Figure 3). Rooftops tend to have irregular shapes in a great number of houses which is a problem for architects because they are being forced to buy excess expensive material to floor them correctly. Some help from an engineer can reduce the cost on the architects and increase their profit. The exact area of an irregular roof can be calculated using integration in order to buy the correct amount of material needed. MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH The area of the roof is determined by dividing it into 4 sections associated with four functions: 𝑓𝑓1 (π‘₯π‘₯), 𝑓𝑓2 (π‘₯π‘₯), 𝑓𝑓3 (π‘₯π‘₯), and 𝑓𝑓4 (π‘₯π‘₯) (see Figure 2). The first three sections present the area which will be floored with artificial grass and the fourth area presents the entrance which will be floored with wood. DOI: Alkhawaldeh: Area of Irregular Shaped Roof 3 Figure 2: Dividing the floor plan into geometric pieces The floor plan shows that the upper border is curve while the lower border is a straight line. The concept of finding area between lines in Calculus is called the area between two curves. Here is its equation: 𝑏𝑏 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = βˆ«π‘Žπ‘Ž [𝑓𝑓(π‘₯π‘₯) βˆ’ 𝑔𝑔(π‘₯π‘₯)] 𝑑𝑑𝑑𝑑, π‘Žπ‘Ž ≀ π‘₯π‘₯ ≀ 𝑏𝑏, (1) where 𝑓𝑓(π‘₯π‘₯) is the function of upper curve while 𝑔𝑔(π‘₯π‘₯) is function of the lower curve. To account for the left and right borders, the integral is bounded between a and b, two points on the curve. Since the second curve on the bottom is a straight line, g(π‘₯π‘₯) = 0 in all the considered cases. Produced by The Berkeley Electronic Press, 2021 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 11, Iss. 2 [2021], Art. 3 4 To find the area of the first three sections, the sections’ functions are plugged in equation (1) as follows: π‘₯π‘₯2 π‘₯π‘₯3 π‘₯π‘₯5 π‘₯π‘₯4 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 (π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) = οΏ½ 𝑓𝑓1 (π‘₯π‘₯)𝑑𝑑𝑑𝑑 + οΏ½ 𝑓𝑓2 (π‘₯π‘₯)𝑑𝑑𝑑𝑑 + οΏ½ 𝑓𝑓3 (π‘₯π‘₯)𝑑𝑑𝑑𝑑 βˆ’ οΏ½ 𝑓𝑓4 (π‘₯π‘₯) 𝑑𝑑𝑑𝑑. (2) π‘₯π‘₯0 π‘₯π‘₯2 π‘₯π‘₯3 π‘₯π‘₯1 The first three sections’ functions: 𝑓𝑓1 (π‘₯π‘₯), 𝑓𝑓2 (π‘₯π‘₯), 𝑓𝑓3 (π‘₯π‘₯) are each put in an integral that is bounded by their limits on the xy-plane and summed together. Function 𝑓𝑓4 (π‘₯π‘₯) is also put in an integral but it is subtracted from the sum since it will be floored with wood instead. After defining the relationship between functions and their areas, we figure out the equations that are presented by the functions. Since the shapes of the sections are rectangular ones with semi-circular ends, the equation of a circle of radius r centered at (h, k) is used: π‘Ÿπ‘Ÿ 2 = (π‘₯π‘₯ βˆ’ β„Ž)2 + (𝑦𝑦 βˆ’ π‘˜π‘˜)2 (3) 𝑦𝑦 = π‘˜π‘˜ Β± οΏ½π‘Ÿπ‘Ÿ 2 βˆ’ (π‘₯π‘₯ βˆ’ β„Ž)2 (4) Rearranging equation (3) in terms of y and restricting it to a semi-circle would result in: The plus minus sign refers to either the upper semi-circle or the lower semi-circle. In sections 1, 3, and 4, it is an upper semi-circle so it is a positive sign, while in section 2 it is a lower semicircle so it is a negative sign. Each function 𝑓𝑓𝑛𝑛 (π‘₯π‘₯) (n=1, 2, 3, 4) can be defined by equation (4) and has three parameters (β„Ž, π‘˜π‘˜, π‘Ÿπ‘Ÿ). Integration of these functions is based on a sin substitution. The values of the four circle centers 𝑐𝑐1 , 𝑐𝑐2 , 𝑐𝑐3 , 𝑐𝑐4 (see Table 1) and the radii π‘Ÿπ‘Ÿ1 , π‘Ÿπ‘Ÿ2 , π‘Ÿπ‘Ÿ3 , π‘Ÿπ‘Ÿ4 (see Table 2) are found DOI: Alkhawaldeh: Area of Irregular Shaped Roof 5 by combining the measures in Figure 1 with the geometric pieces of Figure 2. Since the center of circle 2 is outside the roof, the value of π‘Ÿπ‘Ÿ2 is determined by adding the thickness of the wall, 0.2m (see Figure 1). Plugging in y for 𝑓𝑓1 (π‘₯π‘₯) in equation (2) and plugging in the corresponding values we have: π‘₯π‘₯2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴1 = οΏ½ [π‘˜π‘˜ + οΏ½π‘Ÿπ‘Ÿ 2 βˆ’ (π‘₯π‘₯ βˆ’ β„Ž)2 ]𝑑𝑑𝑑𝑑. π‘₯π‘₯0 (5) Integrating equation (5) we obtain: sin οΏ½2 arcsin οΏ½ (π‘₯π‘₯ βˆ’ β„Ž) π‘₯π‘₯2 π‘Ÿπ‘Ÿ 2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴1 = π‘˜π‘˜π‘˜π‘˜ οΏ½ + οΏ½arcsin οΏ½ οΏ½+ 2 π‘₯π‘₯0 2 π‘Ÿπ‘Ÿ (π‘₯π‘₯ βˆ’ β„Ž) π‘Ÿπ‘Ÿ οΏ½οΏ½ Plugging in the corresponding values results in: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴1 β‰ˆ 120.749 π‘šπ‘š2 Then we use y for 𝑓𝑓2 (π‘₯π‘₯): π‘₯π‘₯3 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴2 = οΏ½ [π‘˜π‘˜ βˆ’ οΏ½π‘Ÿπ‘Ÿ 2 βˆ’ (π‘₯π‘₯ βˆ’ β„Ž)2 ]𝑑𝑑𝑑𝑑. π‘₯π‘₯2 Integrating equation (6) we obtain: Produced by The Berkeley Electronic Press, 2021 (6) οΏ½οΏ½ π‘₯π‘₯2 π‘₯π‘₯0 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 11, Iss. 2 [2021], Art. 3 6 sin οΏ½2 arcsin οΏ½ (π‘₯π‘₯ βˆ’ β„Ž) π‘₯π‘₯3 π‘Ÿπ‘Ÿ 2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴2 = π‘˜π‘˜π‘˜π‘˜ οΏ½ βˆ’ οΏ½arcsin οΏ½ οΏ½+ 2 π‘₯π‘₯2 2 π‘Ÿπ‘Ÿ (π‘₯π‘₯ βˆ’ β„Ž) π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½οΏ½ π‘₯π‘₯3 π‘₯π‘₯2 οΏ½οΏ½ π‘₯π‘₯5 π‘₯π‘₯3 Plugging in the values we have: Using y for 𝑓𝑓3 (π‘₯π‘₯) gives: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴2 β‰ˆ 31.782 π‘šπ‘š2 π‘₯π‘₯5 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴3 = οΏ½ [π‘˜π‘˜ + οΏ½π‘Ÿπ‘Ÿ 2 βˆ’ (π‘₯π‘₯ βˆ’ β„Ž)2 ]𝑑𝑑𝑑𝑑. (7) π‘₯π‘₯3 Integrating equation (7) we obtain: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴3 = π‘˜π‘˜π‘˜π‘˜ οΏ½ 2 (π‘₯π‘₯ βˆ’ β„Ž) π‘₯π‘₯5 π‘Ÿπ‘Ÿ + οΏ½arcsin οΏ½ οΏ½+ π‘₯π‘₯3 2 π‘Ÿπ‘Ÿ sin οΏ½2 arcsin οΏ½ 2 Plugging in the values we get: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴3 β‰ˆ 46.128 π‘šπ‘š2 Finally, using y for 𝑓𝑓4 (π‘₯π‘₯) gives: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴4 = οΏ½ π‘₯π‘₯4 π‘₯π‘₯1 οΏ½π‘Ÿπ‘Ÿ2 βˆ’ (π‘₯π‘₯ βˆ’ β„Ž)2 𝑑𝑑𝑑𝑑 . Integrating equation (8) we have: DOI: (8) (π‘₯π‘₯ βˆ’ β„Ž) π‘Ÿπ‘Ÿ οΏ½οΏ½ Alkhawaldeh: Area of Irregular Shaped Roof 7 sin οΏ½2 arcsin οΏ½ (π‘₯π‘₯ βˆ’ β„Ž) π‘Ÿπ‘Ÿ 2 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴4 = οΏ½arcsin οΏ½ οΏ½+ 2 2 π‘Ÿπ‘Ÿ (π‘₯π‘₯ βˆ’ β„Ž) π‘Ÿπ‘Ÿ οΏ½οΏ½ οΏ½οΏ½ π‘₯π‘₯4 π‘₯π‘₯1 Plugging in the values (note that π‘˜π‘˜ = 0) we obtain: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴4 β‰ˆ 23.363 π‘šπ‘š2 Plugging the equations (5), (6), (7), and (8) into equation (2) reveals that the floor area that will be cover with artificial grass is: 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 (π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴1 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴2 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴3 βˆ’ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴4 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 (π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) = 120.749 + 31.782 + 46.128 βˆ’ 23.363 β‰ˆ 175.3 π‘šπ‘š2 . As calculated previously, the entrance area that will be cover with wood is: β‰ˆ 23.4 π‘šπ‘š2 . DISCUSSION The purpose of this experiment is to find accurate amount and cost of material needed to floor a rooftop. The goal is achieved. Integration helped find the exact area of the roof for the two main parts. The architect will need to buy 175.3 π‘šπ‘š2 of artificial grass to cover the planned grass floor of the roof. Another 23.4 π‘šπ‘š2 of wood is needed to cover the entrance of the roof. Produced by The Berkeley Electronic Press, 2021 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 11, Iss. 2 [2021], Art. 3 8 According to the Jordanian material market, 1 π‘šπ‘š2 of artificial grass costs 11.3 dollars and 1 π‘šπ‘š2 of Parquet wood costs 21.0 dollars (see Table 3). Therefore, the total cost will be 2472 US dollars (175.3 Γ— 11.3 + 23.4 Γ— 21.0). The results are expected since adding the areas together will make a reasonable total area of a roof and the price is within the usual range. Using this technique in the field of design can help architects make their work environmentally efficient. Additionally, materials like artificial grass can be expensive, so this experiment can reduce of amounts paid in excess. CONCLUSION AND RECOMMENDATIONS The area of an irregularly shaped roof is found by building a relationship between different sections that the roof is divided into. Taking the integrals of those sections gives the calculations of the area of the two wanted parts. Suggestions regarding possible future projects revolving around the same issue include building a program that can scan irregular floors and give the needed areas. Another suggestion would be calculating the area of a room with more complicated irregular borders or objects in the middle. DOI: Alkhawaldeh: Area of Irregular Shaped Roof 9 *All Measures are in meters Symbols definition 𝒓𝒓 Radius of a circle 𝒄𝒄 Center of a circle coordinates π’Œπ’Œ Vertical length Horizontal length 𝒉𝒉 REFERENCES Books: 1. Stewart, J. (2016). Calculus: Early transcendentals. Boston, MA, USA: Cengage Learning. 2. Larson, Ron, and Bruce H. Edwards. Calculus of a Single Variable: Early Transcendental Functions. Cengage, 2019. Websites: 1. Circle equation review | Analytic geometry (article). (n.d.). Retrieved December 07, 2020, from 2. Libretexts. (2020, November 17). 1.1: Area Between Two Curves. Retrieved December 07, 2020, from al_Calculus/1:_Area_and_Volume/1.1:_Area_Between_Two_Curves. Produced by The Berkeley Electronic Press, 2021 Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 11, Iss. 2 [2021], Art. 3 10 APPENDIX Table 1: table of points Variable Value (meter) π’„π’„πŸπŸ (3.89, 12.54) π’„π’„πŸ‘πŸ‘ (12.3, 10.4) π’™π’™πŸŽπŸŽ (0, 0) π’„π’„πŸπŸ (8.63, 14.7) π’„π’„πŸ’πŸ’ (8.2, 0) π’™π’™πŸπŸ (4.2, 0.0) π’™π’™πŸπŸ (7.35, 0.0) π’™π’™πŸ‘πŸ‘ (10.22, 0.0) π’™π’™πŸ“πŸ“ (13.8, 0.0) π’™π’™πŸ’πŸ’ (11.9, 0.0) Table 2: table of radii DOI: Variable Value (meter) π’“π’“πŸπŸ 4.47 π’“π’“πŸ‘πŸ‘ 2.72 π’“π’“πŸπŸ 1.71 π’“π’“πŸ’πŸ’ 3.87 Alkhawaldeh: Area of Irregular Shaped Roof 11 Table 3: Material prices in Amman, Jordan Material Artificial grass of 15mm thickness Price (per π’Žπ’ŽπŸπŸ ) Parquet wooden flooring of 11mm thickness $21.0 Figure 3: plan model of the roof garden Produced by The Berkeley Electronic Press, 2021 $11.3
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