Calculus I · Unit 3A · lesson
The Riemann-Sum Definition of the Definite Integral
Learn The Riemann-Sum Definition of the Definite Integral through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Riemann sums and the definite integralWhat this section is building
Learn The Riemann-Sum Definition of the Definite Integral through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.
Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.
Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.
Learning objectives
Explain the definite integral as a limit and translate between sigma and integral notation.
The Riemann-Sum Definition of the Definite Integral
From rectangles to an exact total
A single Riemann sum is an approximation because each rectangle replaces a varying function by one constant height. The definite integral is obtained by considering a sequence of finer partitions whose largest subinterval width approaches zero. If all valid choices of sample points lead to the same limiting value, that value is the integral. The exact total is therefore defined by the stability of increasingly detailed approximations.
This definition explains why the integral is more general than a geometric area formula. The terms may represent signed change, mass, probability, or any other continuously distributed contribution. It also explains why integrability is a mathematical condition rather than a drawing convention: the sums must settle toward one value. Continuous functions on closed intervals satisfy this requirement, giving us a large and reliable class of integrable functions.
Definite integral
If the limit exists and is independent of the sample points,
The integral is one number. The Riemann sums are approximations indexed by . As the largest subinterval width approaches zero, the approximations converge to the integral.
Evaluate from the definition
Use right endpoints with and :
Taking the limit gives
Integrability is a theorem, not a visual promise
Continuous functions on closed intervals are Riemann integrable. Many discontinuous functions are also integrable, but not every bounded function is. A later analysis course makes precise how the set and severity of discontinuities control Riemann integrability.
u3a-riemann-limit-01Evaluate .
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Divide numerator and denominator by .
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Rectangles converging to a fixed total. A parabola with increasingly narrow rectangles and a numerical estimate approaching 8/3. Show Riemann rectangles becoming narrower while the sum stabilizes. Avoid saying rectangles become area; state their summed areas converge to the integral.
Every relationship in rectangles converging to a fixed total uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show Riemann rectangles becoming narrower while the sum stabilizes.
Rectangles converging to a fixed total. Show Riemann rectangles becoming narrower while the sum stabilizes.
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