Solving problems involving numerical integration (I): Incorporating different techniques

William Guo; William Guo

doi:10.3934/steme.2023009

STEM Education

2023, Volume 3, Issue 2: 130-147. doi: 10.3934/steme.2023009

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Classroom note

Solving problems involving numerical integration (I): Incorporating different techniques

William Guo ^,

School of Engineering and Technology, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4702, Australia

Academic Editor: Dianshun Hu

Received: 10 May 2023 Revised: 26 June 2023 Accepted: 30 June 2023 Published: 05 July 2023

Numerical integration plays an important role in solving various engineering and scientific problems and is often learnt in applied calculus commonly through the trapezium and Simpson's methods (or rules). A common misconception for some students is that Simpson's method is the default choice for numerical integration due to its higher accuracy in approximation over the trapezium method by overlooking the requirement for using Simpson's method. As learning progressed to other numerical methods scheduled later in advanced mathematics, such as interpolations and computational modelling using computing tools like MATLAB, there is a lack of articulation among these numerical methods for students to solve problems solvable only by combining two or more approaches. This classroom note shares a few teaching and learning practices the author experienced in lectures, tutorials, and formal assessments on comparing or combining different numerical methods for numerical integration for engineering students in applied calculus and advanced mathematics over the past decade at Central Queensland University (CQU), a regional university in Australia. Each case represents a common concern raised or a mistake made by some students in different times. These efforts helped not only correct the misconception on the use of Simpson's method by some students, but also develop students' strategic thinking in problem solving, particularly involving decision-making for choosing the best possible method to produce a more appropriate solution to a problem that does not have an analytical solution.
- numerical methods,
- numerical integration,
- trapezium method,
- Simpson's method,
- error bound,
- interpolation,
- Maclaurin series,
- computing tools
Citation: William Guo. Solving problems involving numerical integration (I): Incorporating different techniques[J]. STEM Education, 2023, 3(2): 130-147. doi: 10.3934/steme.2023009

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Abstract

Numerical integration plays an important role in solving various engineering and scientific problems and is often learnt in applied calculus commonly through the trapezium and Simpson's methods (or rules). A common misconception for some students is that Simpson's method is the default choice for numerical integration due to its higher accuracy in approximation over the trapezium method by overlooking the requirement for using Simpson's method. As learning progressed to other numerical methods scheduled later in advanced mathematics, such as interpolations and computational modelling using computing tools like MATLAB, there is a lack of articulation among these numerical methods for students to solve problems solvable only by combining two or more approaches. This classroom note shares a few teaching and learning practices the author experienced in lectures, tutorials, and formal assessments on comparing or combining different numerical methods for numerical integration for engineering students in applied calculus and advanced mathematics over the past decade at Central Queensland University (CQU), a regional university in Australia. Each case represents a common concern raised or a mistake made by some students in different times. These efforts helped not only correct the misconception on the use of Simpson's method by some students, but also develop students' strategic thinking in problem solving, particularly involving decision-making for choosing the best possible method to produce a more appropriate solution to a problem that does not have an analytical solution.

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