Integration as the limit of rectangle sums
| English | 中文 | Pinyin |
|---|---|---|
| Riemann sum/ˈriːmən sʌm/ | 黎曼和 | lí màn hé |
A finite collection of rectangles misses part of a curved region. What changes as the strip width tends to zero?
- A finite collection of rectangles misses part of a curved region. What changes as the strip width tends to zero?
- This lesson studies Riemann sum 黎曼和: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
Choose the mathematical structure
- For n equal strips on [a,b], width h=(b−a)/n. A right-end sum is h times the sum of f(a+kh) for k=1,…,n; a left-end sum uses k=0,…,n−1. For continuous f, both limits as n tends to infinity equal the integral. A finite sum is an approximation; the limit is the exact signed accumulation.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines Riemann sum?
A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For f(x)=x² on [0,1], h=1/n. The right sum R_n=(1/n³) sum from k=1 to n of k²=(n+1)(2n+1)/(6n²)=1/3+1/(2n)+1/(6n²). The left sum L_n=(n−1)n(2n−1)/(6n³)=1/3−1/(2n)+1/(6n²). With n=2, L_2=1/8 and R_2=5/8; both bound the exact integral 1/3 because x² increases on this interval. Their difference is 1/n, tending to zero. This same definition gives signed contributions: for f=−2 on [0,1], every such sum is −2, while positive geometric area is 2.
Integration as the limit of rectangle sums
For n equal strips on [a,b], width h=(b−a)/n
Check the hypothesis, endpoints and coefficients behind each integral calculation.
Find L_2 for x² on [0,1].
The left heights 0 and 1/4, each of width 1/2, give 1/8.
Test a tempting shortcut
- Multiply by strip width: adding heights alone gives the wrong units and limit. The left sum includes k=0 and excludes k=n; the right sum does the reverse. Increasing functions give left/right bounds, but the rule must be checked on the actual interval. Rectangle sums and the trapezium rule are different finite approximations. Negative heights contribute negative signed area.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A finite right-end rectangle sum is already the exact integral of every continuous curve. This claim is false. Explain which definition or assumption it violates.
Find R_2.
The right heights 1/4 and 1 give (1/2)(5/4)=5/8.
A finite right-end rectangle sum is already the exact integral of every continuous curve.
Multiply by strip width: adding heights alone gives the wrong units and limit. The left sum includes k=0 and excludes k=n; the right sum does the reverse. Increasing functions give left/right bounds, but the rule must be checked on the actual interval. Rectangle sums and the trapezium rule are different finite approximations. Negative heights contribute negative signed area.
Interpret a new situation
- Write the partition, evaluation points and sigma limits before simplifying. Use a finite-sum identity, then evaluate the limit. Compare finite bounds with the antiderivative result as an independent check. This is H4 exact limit reasoning; numerical trapezia belong to I3 and are not a substitute for the limiting argument.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the common limit of L_n and R_n.
Terms 1/(2n) and 1/(6n²) tend to zero, leaving 1/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · H. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A sum of signed rectangle contributions that approaches an integral as the partition becomes finer. Choose the relationship, show the method, check its assumptions and interpret the result.