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Visual guide · Calculus I · Unit 3A

Riemann Sum Progression: From Rectangles to Definite Integral

A visual progression from a coarse partition to a definite integral, with width, height, and limiting behavior labeled.

Reference format12 min estimated timeFoundational progression

What is included

See how increasingly fine Riemann sums approximate signed accumulation and lead to a definite integral.

Skills assessed

  • interpreting Riemann sums
  • connecting sums to integrals

Prerequisites

  • function graphs
  • sigma notation
Riemann Sum Progression instructional sequence
A visual progression from a coarse partition to a definite integral, with width, height, and limiting behavior labeled. The numbered labels and written sequence preserve meaning without relying on color.
Long description

Read the diagram from top to bottom. Each numbered box names one decision or mathematical operation. Arrows show the required order; the text labels remain the complete interpretation in print, dark mode, and nonvisual reading.

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Student PDF (opens in a new tab)SVGPNG

Reference formulas

Accumulation

Δx=(b−a)/n\Delta x=(b-a)/n∑i=1nf(xi∗)Δx\sum_{i=1}^n f(x_i^*)\Delta xlim⁡n→∞∑f(xi∗)Δx=∫abf(x) dx\lim_{n\to\infty}\sum f(x_i^*)\Delta x=\int_a^b f(x)\,dx

Common errors

  • Treating rectangle height as area or forgetting signed contribution below the axis.

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Supporting steps