Integral calculus - Area AASL5.11
Keywords
[table][br][br][tr][br][td]Integration[/td][br][td]積分[/td][br][td]적분[/td][br][td]积分[/td][br][/tr][br][tr][br][td]Area under the curve[/td][br][td]曲線の下の面積[/td][br][td]곡선 아래 면적[/td][br][td]曲线下的面积[/td][br][/tr][br][tr][br][td]Definite integral[/td][br][td]定積分[/td][br][td]정적분[/td][br][td]定积分[/td][br][/tr][br][tr][br][td]Trapezoidal rule[/td][br][td]台形則[/td][br][td]사다리꼴 규칙[/td][br][td]梯形规则[/td][br][/tr][br][tr][br][td]Geometric shapes[/td][br][td]幾何学的形状[/td][br][td]기하학적 도형[/td][br][td]几何形状[/td][br][/tr][br][tr][br][td]Function[/td][br][td]関数[/td][br][td]함수[/td][br][td]函数[/td][br][/tr][br][tr][br][td]Algebraic method[/td][br][td]代数的方法[/td][br][td]대수적 방법[/td][br][td]代数方法[/td][br][/tr][br][tr][br][td]Area between curves[/td][br][td]曲線間の面積[/td][br][td]곡선 사이의 면적[/td][br][td]曲线之间的面积[/td][br][/tr][br][tr][br][td]Numerical methods[/td][br][td]数値方法[/td][br][td]수치 방법[/td][br][td]数值方法[/td][br][/tr][br][tr][br][td]Analytical methods[/td][br][td]解析的方法[/td][br][td]해석적 방법[/td][br][td]分析方法[/td][br][/tr][br][tr][br][td]Precision[/td][br][td]精度[/td][br][td]정밀도[/td][br][td]精确度[/td][br][/tr][br][tr][br][td]Estimation[/td][br][td]推定[/td][br][td]추정[/td][br][td]估计[/td][br][/tr][br][/table][br]
[table][br][tr][br][td][b]Factual Inquiry[/b][/td][br][td][b]Conceptual Inquiry[/b][/td][br][td][b]Debatable Inquiry[/b][/td][br][/tr][br][tr][br][td]1. How does the process of integration relate to finding the area under a curve?[/td][br][td]1. What conceptual relationship exists between the algebraic method of integration and the geometric approach of area calculation?[/td][br][td]1. To what extent does the precision of integration as a mathematical tool affect its application in real-world problems?[/td][br][/tr][br][tr][br][td]2. In what ways do the geometric properties of shapes, like triangles and trapezoids, assist in estimating the area under a curve?[/td][br][td][/td][br][td]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?[/td][br][/tr][br][/table][br]
For the function f(x)=x Move point A to (0,0) and point B to (10,10)[br][br]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).[br]Calculate the area of the trapezium formed. [br]Compare this to the integral.[br]Comment on your findings.
Move point A to (-10,-10) and point B to (0,0)[br][br]Confirm the area of the triangle remains the same but integral is now -50.[br]How does the area under/above the curve relate to the integral.
Move point to (10,10)[br][br]The area of the two triangles is now, [math]50units^2+50units^2=100units^2[/math] whilst the area of the integral is [math]0[/math]. [br][br]How can you explain this. Compare it to [math]\text{\int|f(x)| dx}[/math].[br][br]Experiment with moving points A and B. [br][br]When is the [math]\text{\int|f(x)| dx}[/math] equal to [math]\text{\int f(x) dx}[/math]
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.[br][br]What can we conjecture about the relationship between [math]\int_a^bf(x)dx[/math] and[math]\int_b^af(x)dx[/math]
Part 2 - Area between curves
Here we will look at working areas enclosed between two functions
Function Positioning: Observe the graph and identify which function, [math]f(x)[/math] or [math]g(x)[/math], lies above the other between points A and B. Which function should be subtracted from which to find the area between them?[br][br]Swap the functions, so [math]\text{f(x)=x^3+2}[/math], and [math]g(x)=4x-1[/math]. How does this change the calculation?
Experiment with using the slider to translate the curves upwards. [br][br]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.
Part 2 - Practice questions
[b]Selection of questions from Christos.[/b][br][br]Question 1-4 Practice questions[br]Questions 5-34 Section A - Short answer style questions[br]Question 34-60. Section B - Long answer style questions
[MAA 5.11] DEFINITE INTEGRALS - AREAS
[MAA 5.11] DEFINITE INTEGRALS - AREAS_solutions
Part 3 - Support materials
Videos on this.
Areas using GDC
Areas without GDC
Areas between curves
Lesson plan
