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Riemann Sums — Syllabus Points

1. Upper and Lower Riemann Sums​

  • 1.1 Partition interval [a,b][a,b] into nn equal subintervals of width Δx=b−an\Delta x = \frac{b-a}{n}
  • 1.2 For an increasing function: upper sum uses right endpoints, lower sum uses left endpoints
  • 1.3 For a decreasing function: upper sum uses left endpoints, lower sum uses right endpoints
  • 1.4 Express sums in sigma notation and evaluate using standard summation formulae

2. Standard Summation Formulae​

  • 2.1 ∑r=1nr=n(n+1)2\sum_{r=1}^n r = \frac{n(n+1)}{2}
  • 2.2 ∑r=1nr2=n(n+1)(2n+1)6\sum_{r=1}^n r^2 = \frac{n(n+1)(2n+1)}{6}
  • 2.3 ∑r=1nr3=n2(n+1)24\sum_{r=1}^n r^3 = \frac{n^2(n+1)^2}{4}
  • 2.4 ∑r=1n1=n\sum_{r=1}^n 1 = n

3. Stirling-Type Approximations​

  • 3.1 Express ln⁡N!=∑r=1Nln⁡r\ln N! = \sum_{r=1}^N \ln r
  • 3.2 Bound ln⁡N!\ln N! between integrals: ∫1Nln⁡x dx≤ln⁡N!≤∫1Nln⁡x dx+ln⁡N\int_1^N \ln x\,dx \le \ln N! \le \int_1^N \ln x\,dx + \ln N
  • 3.3 Use Riemann sums to refine Stirling's approximation
  • 3.4 Relate sums of ln⁡r\ln r to ∫ln⁡x dx=xln⁡x−x+C\int \ln x\,dx = x\ln x - x + C

4. Limit of Riemann Sums​

  • 4.1 Express definite integrals as limits of Riemann sums
  • 4.2 Convert between sum notation and definite integral
  • 4.3 Apply the limit n→∞n\to\infty to obtain exact integral values