Riemann sums
Objective. Construct definite integrals from limiting sums.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A Riemann sum approximates the area under a curve y=f(x) on [a,b] by partitioning the interval into n subintervals of width Δx=(b−a)/n and summing f(xᵢ*)Δx over sample points xᵢ*, giving Σf(xᵢ*)Δx. As n→∞ and Δx→0, this sum converges to the definite integral ∫[a,b] f(x)dx, which is how integrals are formally defined as limits of sums. For f(x)=x on [0,1] with right endpoints, the sum is Σ(i/n)(1/n) for i=1 to n, which simplifies to (n+1)/(2n), tending to 1/2 as n→∞, matching ∫₀¹x dx=1/2. Left, right, and midpoint Riemann sums differ in sample-point choice but all converge to the same integral for continuous f.
Worked example
Using a right-endpoint Riemann sum with n=4 equal subintervals, approximate ∫₀⁴ x dx.
- 1Subinterval width Δx = (4−0)/4 = 1, with right endpoints x=1,2,3,4.
- 2Evaluate f(x)=x at each: f(1)=1, f(2)=2, f(3)=3, f(4)=4.
- 3Sum the values: 1+2+3+4 = 10, then multiply by Δx=1.
- 4The Riemann sum approximation is 10, close to the exact value ∫₀⁴x dx=8.
Answer: 10
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Start practisingCommon mistake. Students assume left and right Riemann sums always give the same numeric value for finite n; in fact they generally differ unless f is constant, and only their common limit as n→∞ equals the exact integral.
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