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Math10B Lecture 4 Notes

Dec 8, 2025

Overview

  • Defines the definite integral as the limit of Riemann sums.
  • Explains Riemann sums (generalization of left/right/midpoint sums).
  • States conditions for integrability and geometric interpretation as signed area.
  • Works examples computing definite integrals and total area via |f|.

Riemann Sums (Definition)

  • Divide [a, b] into N equal subintervals of width Δx = (b - a) / N.
  • In each subinterval choose any sample point xk (left/right/midpoint or any).
  • Riemann sum: R_N = Σ_{k=1}^N f(x_k) Δx.
  • Left and right sums are special cases of Riemann sums._

Definite Integral (Definition and Notation)

  • If lim_{N→∞} R_N exists, define the definite integral: ∫a^b f(x) dx = lim{N→∞} Σ_{k=1}^N f(x_k) Δx.
  • Integral symbol ∫ resembles an elongated "S" (sum).
  • Terminology:
    • a = lower limit of integration.
    • b = upper limit of integration.
    • f(x) = integrand.
    • dx = differential, evokes Δx (infinitesimal change in x).

Integrability (Theorems / Conditions)

  • If f is continuous on [a, b], then f is integrable on [a, b].
  • More generally, f is integrable if it has only finitely many jump discontinuities on [a, b].
  • If the limit of Riemann sums does not exist, f is not integrable on that interval.

Geometric Interpretation: Signed Area

  • If f(x) ≥ 0 on [a, b], then ∫_a^b f(x) dx equals the area under the curve above the x-axis between x=a and x=b.
  • If f(x) ≤ 0 on [a, b], each rectangle in Riemann sums has negative height; the integral equals the negative of the area below the x-axis.
  • For functions that change sign, the definite integral equals: (area above x-axis) − (area below x-axis).
  • Thus the integral gives signed area, not total geometric area._

Key Terms and Definitions

  • Riemann Sum: Σ_{k=1}^N f(x_k) Δx using one sample point per subinterval.
  • Δx: Width of each subinterval = (b − a) / N.
  • Integrand: The function f(x) being integrated.
  • Differential (dx): Represents an infinitesimal change in x, analog of Δx.
  • Definite Integral: ∫_a^b f(x) dx = limit of Riemann sums as N→∞.
  • Integrable: A function for which the definite integral exists on [a, b].
  • Signed Area: Integral value that counts area above x-axis positive, below negative.

Example: Signed Integral Calculation

  • Given continuous f on [0,7] that is sometimes positive, sometimes negative.
  • ∫_0^7 f(x) dx = (sum of areas above x-axis) − (sum of areas below x-axis).
  • In the prepared example:
    • Areas partition into triangular regions with bases and heights of 2, 2, 2, and 1.
    • Areas: above = 2 + 1 = 3; below = 1/2 + 2 = 2.5.
    • Integral ∫_0^7 f(x) dx = 3 − 2.5 = 0.5 = 1/2.

Example: Total Geometric Area (Using |f|)

  • Total area bounded by curve and x-axis equals ∫_a^b |f(x)| dx.
  • For the same example:
    • Sum absolute areas: 1/2 + 2 + 2 + 1 = 11/2 = 5.5.
    • This equals ∫_0^7 |f(x)| dx, not ∫0^7 f(x) dx.

Action Items / Next Steps (If Studying)

  • Practice computing Riemann sums for simple functions using left/right/midpoint choices.
  • Evaluate definite integrals by interpreting signed areas from graphs.
  • Compute total geometric area by integrating |f(x)| when function changes sign.
  • Verify integrability by checking continuity or identifying only finitely many jump discontinuities.

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