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Home›Calculus›Integrals

Approximating area with rectangles

Four rectangles under y = x² from 0 to 2 give 1.75 if you use left ends and 3.75 if you use right ends. The true area, 8/3, lies in between.

Shapes with known area

Rectangles, triangles and trapezoids have area formulas; a region with a curved edge does not. OpenStax starts the integration chapter with Archimedes, who filled curved regions with smaller and smaller shapes of known area and added them up, a process now called the method of exhaustion. Calculus does the same with rectangles. Cut the interval [a, b] into n pieces of equal width Δx = (b − a)/n, stand a rectangle on each piece with its height taken from the curve, and add the areas. More, thinner rectangles fit the curve better.

Sigma notation

Adding many terms calls for short notation. The capital Greek letter sigma writes a sum compactly: Σi=120 i means 1 + 2 + ⋯ + 20. The letter i is the index; it starts at the value below the sigma and stops at the value above. OpenStax's first example is Σi=15 3i = 3 + 9 + 27 + 81 + 243 = 363. Two formulas save time later: Σi=1n i = n(n + 1)/2 and Σi=1n i² = n(n + 1)(2n + 1)/6. So 1 + 2 + ⋯ + 20 = 20 × 21/2 = 210.

Left and right endpoints

Each rectangle needs a height, and the simplest choice is the curve's value at one end of its piece. Using left ends gives the left-endpoint approximation Ln = Σ f(xi−1)Δx; using right ends gives Rn = Σ f(xi)Δx. For f(x) = x² on [0, 2] with n = 4, Δx = 0.5. The left heights are f(0), f(0.5), f(1), f(1.5) = 0, 0.25, 1, 2.25, so L4 = 0.5 × 3.5 = 1.75. The right heights are 0.25, 1, 2.25, 4, so R4 = 0.5 × 7.5 = 3.75. These are OpenStax's numbers.

L4 for y = x² on [0, 2]: rectangles of width 0.5 with left-end heights 0, 0.25, 1 and 2.25. Their total, 1.75, misses the area above them.

Because x² is increasing on [0, 2], each left rectangle sits under the curve and each right rectangle pokes above it. So L4 underestimates and R4 overestimates, and the true area is between 1.75 and 3.75. For a decreasing function the roles swap. Doubling n to 8 gives L8 = 2.1875 and R8 = 3.1875, a narrower bracket; at n = 32 the bracket is 2.54 to 2.79. The exact area, which unit 4 computes in one line, is 8/3 ≈ 2.667.

BackgroundArchimedes, working in the third century BCE, used this kind of argument to find the area of a parabolic segment and to bound π between two polygons. He never had a general symbol for a sum or a limit; each result was a separate geometric proof. The idea of adding up thin strips waited about two thousand years for the notation that made it routine.
Write the partition down before adding anything. For n = 4 on [0, 2] the cut points are x0 = 0, x1 = 0.5, x2 = 1, x3 = 1.5 and x4 = 2. Left sums use x0 to x3, right sums use x1 to x4.
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y = x² on [0, 2] with n = 4 left rectangles. Switch to Right, then press Double n a few times and watch Ln and Rn close in on 8/3 ≈ 2.667.
  1. Σi=120 i = ?
  2. Σi=15 3i = ?
  3. Σi=36 2i = ?
  4. Σi=110 i² = ?
  5. [1, 3] split into n = 8 equal pieces. Δx = ?
  6. L4 for x² on [0, 2]?
  7. R4 for x² on [0, 2]?
  8. L6 for x²/2 on [0, 3] (OpenStax figure 5.5)?
  9. R6 for x²/2 on [0, 3]?
  10. f is increasing on [a, b]. The left-endpoint sum is:
    • an underestimate
    • an overestimate
    • exact
    • negative
Key points
  • Δx = (b − a)/n; n rectangles of width Δx.
  • Σ notation: Σi=1n i = n(n + 1)/2, Σ i² = n(n + 1)(2n + 1)/6.
  • Ln uses left ends, Rn right ends.
  • For an increasing f, Ln < area < Rn; more rectangles narrow the gap.
Sources for this course (6)
  • OpenStax, Calculus Volume 1 , chapter 5 Integration: 5.1 Approximating Areas (Archimedes and the method of exhaustion; sigma notation with Σ 3^i for i = 1 to 5 = 363; left and right endpoint approximations L4 = 1.75 and R4 = 3.75 for x² on [0, 2], L6 = 3.4375 and R6 = 5.6875 for x²/2 on [0, 3]; (x − 1)³ + 4 on [0, 2] approaching 8; Riemann sums named for the 19th-century mathematician Bernhard Riemann; upper and lower sums: openstax.org/books/calculus-volume-1/pages/5-1-approximating-areas)
  • 5.2 The Definite Integral (limit of Riemann sums, integrability of continuous functions, net signed area, properties: .../pages/5-2-the-definite-integral)
  • 5.3 The Fundamental Theorem of Calculus (Newton and Leibniz, among others, late 1600s and early 1700s; mean value theorem for integrals; parts 1 and 2: .../pages/5-3-the-fundamental-theorem-of-calculus)
  • 5.4 Integration Formulas and the Net Change Theorem (.../pages/5-4-integration-formulas-and-the-net-change-theorem)
  • 4.10 Antiderivatives (.../pages/4-10-antiderivatives)
  • MacTutor, University of St Andrews: Archimedes (287 BC to 212 BC; area of a parabolic segment; π bounded with 96-sided inscribed and circumscribed polygons: mathshistory.st-andrews.ac.uk/Biographies/Archimedes) and Leibniz (first used the ∫ f(x) dx notation in a manuscript of 21 November 1675: mathshistory.st-andrews.ac.uk/Biographies/Leibniz).

Checked: 2026-10-17