The key concept in numerical integration is the Riemann Sum. The area beneath the curve is divided into small rectangular strips whose area is found and summed. This sum will approximate the integral. However it will overestimate or underestimate. The trapezium rule uses trapezia instead of rectangles. Numerical integration gives an approximation of the integral. We can find upper and lower bounds of the actual integral using this approximation. Common misconceptions or difficulties learners may have: Common difficulties with numerical methods to find roots include lack of familiarity with the algorithms; identification of convergence; not enough detail in the iterates; not knowing how to proceed after failure; not taking time to check that the root approximation is genuine. Learners also find the concept of divergence difficult, and may find it hard to accept that numerical methods is part of applied mathematics and so at times will fail. It is a misconception that roots will always be obtained using these methods. Common difficulties with simple iterative methods are the staircase and cobweb diagrams. Students need to practise drawing these and they need time to comprehend how the algebra fits with the diagram and the root approximation. Common difficulties with Newton-Raphson are sometimes met obtaining the derivative; understanding the derivation of the formula; sketching the iteration diagram and becoming familiar with failure cases. Numerical integration in practice is fairly straightforward but are conceptually Riemann’s ideas challenging.
The key concept in numerical integration is the Riemann Sum. The area beneath the curve is divided into small rectangular strips whose area is found and summed. This sum will approximate the integral. However it will overestimate or underestimate. The trapezium rule uses trapezia instead of rectangles. Numerical integration gives an approximation of the integral. We can find upper and lower bounds of the actual integral using this approximation. Common misconceptions or difficulties learners may have: Common difficulties with numerical methods to find roots include lack of familiarity with the algorithms; identification of convergence; not enough detail in the iterates; not knowing how to proceed after failure; not taking time to check that the root approximation is genuine. Learners also find the concept of divergence difficult, and may find it hard to accept that numerical methods is part of applied mathematics and so at times will fail. It is a misconception that roots will always be obtained using these methods. Common difficulties with simple iterative methods are the staircase and cobweb diagrams. Students need to practise drawing these and they need time to comprehend how the algebra fits with the diagram and the root approximation. Common difficulties with Newton-Raphson are sometimes met obtaining the derivative; understanding the derivation of the formula; sketching the iteration diagram and becoming familiar with failure cases. Numerical integration in practice is fairly straightforward but Riemann’s ideas are conceptually challenging.