Integral calculus - Area AASL5.11
Keywords
| Integration | 積分 | 적분 | 积分 |
| Area under the curve | 曲線の下の面積 | 곡선 아래 면적 | 曲线下的面积 |
| Definite integral | 定積分 | 정적분 | 定积分 |
| Trapezoidal rule | 台形則 | 사다리꼴 규칙 | 梯形规则 |
| Geometric shapes | 幾何学的形状 | 기하학적 도형 | 几何形状 |
| Function | 関数 | 함수 | 函数 |
| Algebraic method | 代数的方法 | 대수적 방법 | 代数方法 |
| Area between curves | 曲線間の面積 | 곡선 사이의 면적 | 曲线之间的面积 |
| Numerical methods | 数値方法 | 수치 방법 | 数值方法 |
| Analytical methods | 解析的方法 | 해석적 방법 | 分析方法 |
| Precision | 精度 | 정밀도 | 精确度 |
| Estimation | 推定 | 추정 | 估计 |
| Factual Inquiry | Conceptual Inquiry | Debatable Inquiry |
| 1. How does the process of integration relate to finding the area under a curve? | 1. What conceptual relationship exists between the algebraic method of integration and the geometric approach of area calculation? | 1. To what extent does the precision of integration as a mathematical tool affect its application in real-world problems? |
| 2. In what ways do the geometric properties of shapes, like triangles and trapezoids, assist in estimating the area under a curve? | 2. Can the method of integration be seen as a universal tool across different branches of science, or are its limitations a barrier to its application in certain areas? |
For the function f(x)=x Move point A to (0,0) and point B to (10,10) Calculate the area of the triangle. Compare this to the integral. Comment on your findings.
Move point A to (5,5) and point B to (15,15). Calculate the area of the trapezium formed. Compare this to the integral. Comment on your findings.
Move point A to (-10,-10) and point B to (0,0) Confirm the area of the triangle remains the same but integral is now -50. How does the area under/above the curve relate to the integral.
Move point to (10,10) The area of the two triangles is now, whilst the area of the integral is . How can you explain this. Compare it to . Experiment with moving points A and B. When is the equal to
Optional
Move point A to (0,0) and point B to (10,10). Note the integral.Move point B to (10,10) and point A to (0,0). Note the integral. What can we conjecture about the relationship between and
Part 2 - Area between curves
Function Positioning: Observe the graph and identify which function, or , lies above the other between points A and B. Which function should be subtracted from which to find the area between them? Swap the functions, so , and . How does this change the calculation?
Experiment with using the slider to translate the curves upwards. How does this simplfiy the overall calculation and explain why we don't need to consider when the area enclosed between the curves lie above or below the x-axis.
![[MAA 5.11] DEFINITE INTEGRALS - AREAS.pdf](https://www.geogebra.org/resource/ed8tkdqr/KUYPCUCBv7UGU11y/material-ed8tkdqr-thumb.png)
![[MAA 5.11] DEFINITE INTEGRALS - AREAS_solutions.pdf](https://www.geogebra.org/resource/m33xucpn/fKYXShwQhuLGr5Z4/material-m33xucpn-thumb.png)
