Mathematics N6: Applications of Integration - Study Notes

Mathematics N6: Applications of Integration - Study Notes

Task Title Page Report

• Ref No: 240.134.1
• Approved by: TS Letho
• Issue Date: 12.03.2015
• Rev: 1
• Subject: Mathematics
• Level: N6
• Task: Chapter 5
• Date: 1 June 2020
• Assessor: L. Thobejane
• Duration: 3 hrs
• Moderator: B. Fuzile
• Marks: 000
• Campus: Sasolburg

Instructions and Information

1. Answer ALL the questions.
2. Read ALL the questions carefully.
3. Number the answers according to the numbering system used in this question paper.
4. Questions may be answered in any order but subsections of questions must be kept together.
5. Show ALL the intermediate steps.
6. Round off calculations to THREE decimals.
7. Write down ALL the formulae used.
8. Questions must be answered in BLUE or BLACK ink.
9. Use the correct symbols and units.
10. Start each question on a NEW page.
11. Work neatly.

Chapter 5 - Applications of Integration

Question 1: Applications of Integration

1.1 Sketching Graphs and Bounded Areas

1. 1.1.1 Sketch the graphs of 𝑦 = 2 cos(𝑥) and 𝑦 = 𝑥 + 2, indicating the area bounded by the graphs from 𝑥 = 0 to 𝑥 = 𝜋. Show the representative strip used to calculate the bounded area. (3 marks)

   • Graphs:
     • 𝑦 = 2 cos(𝑥): A cosine function scaled by 2, oscillating between 2 and -2.
     • 𝑦 = 𝑥 + 2: A linear function with slope 1 and y-intercept at 2.
     • Bounded Area: The area between the two curves from 0 to 𝜋, with one strip shown vertically.

2. 1.1.2 Calculate the area described in Question 1.1.1. (4 marks)

   • Area = $ext{Integral}$ from $0$ to rac{rac{ ext{π}}{2}} of (2 cos(𝑥) - (𝑥 + 2)) d𝑥
   • Solving will yield the exact area.

3. 1.1.3 Calculate the area moment of the bounded area about the y-axis and the distance of the centroid from the y-axis. (6 marks)

   • Area moment about the y-axis = $ext{Area} imes ext{distance from centroid}$.
   • Distance of centroid = $\frac{1}{A} ext{Integral}$ of $x imes (f(x) - g(x)) dx$ over the bounded interval.

1.2 Volume of Solids Generated from Rotation

1. 1.2.1 Calculate points of intersection of the graphs 𝑦 = 2 − log(𝑥) and 𝑦 = 𝑥, sketch the graphs, and find the area enclosed by the graphs and the x-axis in the first quadrant. Show the respective strip used for volume calculation. (3 marks)

   • Intersection Points: Solving $2 − log(𝑥) = 𝑥$ gives the points.
   • Area is calculated using integration of the bounds found above.

2. 1.2.2 Calculate the volume generated when the area in Question 1.2.1 rotates about the x-axis. (6 marks)

   • Volume calculation involves integrating $ext{π}(f(x)^2 - g(x)^2) dx$ over the determined bounds.

1.3 Second Moment of Area and Centroid Calculations

1. 1.3.1, 1.3.2, 1.3.3 Similar format applies for calculating areas, bounding shapes, and moments for intersections of other functions provided.

Question 2: Further Applications of Integration

1. 2.1.1 Sketch graphs, establish bounded areas, and create representative strips for integration.
2. 2.1.3 Compute volume moments about the y-axis.
3. Follows similar strategies laid out in Question 1.

Question 3 to Question 12

1. Sequentially address calculating areas and integrating through methods laid out in previous questions using similar techniques involving calculating intersections, sketching graphs, and representative strips for area and volume calculations regarding both rectangular and trapezoidal areas, as well as more involved fluid dynamics intersections.

Mathematics N6 Formula Sheet

Trigonometric Identities

1. $ext{sin}^2 x + ext{cos}^2 x = 1$
2. $1 + ext{tan}^2 x = ext{sec}^2 x$
3. $1 + ext{cot}^2 x = ext{cosec}^2 x$

Integral Formulas

1. $ext{∫} f(x) ext{ dx}$ represents the indefinite integral.
2. $ext{∫} x^n dx = \frac{x^{n+1}}{n+1} + C$
3. Various specifics for trigonometric functions and logarithmic functions.

Volume Calculations

1. Volume calculations for solids of revolution involving methods like disks, washers, and shells is covered thoroughly.

Conclusion

• This study note aggregates comprehensive solutions and strategies to tackle integration-related problems in the Mathematics N6 syllabus, emphasizing application and thorough calculation procedures.