Integration as the limit of rectangle sums
Введено| English | Значение |
|---|---|
| Riemann sum/ˈriːmən sʌm/ | сумма Римана |
What changes as the strip width tends to zero?
- A finite collection of rectangles misses part of a curved region.
- This lesson studies Riemann sum 黎曼和: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
The method
- 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.
For x² on [0,1], L₂=1/8 and R₂=5/8. Why do these bound the exact integral?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
On an increasing function each rectangle’s left height is the strip minimum and its right height the maximum, trapping the true area. The gap R_n-L_n=1/n→0 forces both limits to 1/3.
A worked case
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].
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The left heights 0 and 1/4, each of width 1/2, give 1/8.
Where it goes wrong
- 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.
Not true: A finite right-end rectangle sum is already the exact integral of every continuous curve. A finite rectangle sum is an approximation; it becomes the integral only in the limit.
Find R_2.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
A finite rectangle sum is an approximation; it becomes the integral only in the limit.
Applying it
- 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.
Find the common limit of L_n and R_n. Give it to 3 significant figures.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Terms 1/(2n) and 1/(6n²) tend to zero, leaving 1/3.
Match each Riemann-sum feature to its rule.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Heights without the width factor carry the wrong units and the wrong limit; signed heights give signed contributions.
In your course
- AQA A-level Mathematics (7357), specification section H.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Riemann sum: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
- For n equal strips on [a,b], width h=(b−a)/n.
- Watch for: multiply by strip width: adding heights alone gives the wrong units and limit.