Calculus I · Unit 3A · lesson

The Riemann-Sum Definition of the Definite Integral

Concept

Learning objectives

Explain the definite integral as a limit and translate between sigma and integral notation.

The Riemann-Sum Definition of the Definite Integral

Explanation

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.

Definition

Definite integral

If the limit exists and is independent of the sample points,

\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i^*)\Delta x.

The integral is one number. The Riemann sums are approximations indexed by $n$ . As the largest subinterval width approaches zero, the approximations converge to the integral.

Worked example

Evaluate $\int_0^1 x\,dx$ from the definition

Use right endpoints with $\Delta x=1/n$ and $x_i=i/n$ :

\sum_{i=1}^n \frac{i}{n}\frac1n =\frac1{n^2}\sum_{i=1}^n i =\frac1{n^2}\frac{n(n+1)}2 =\frac{n+1}{2n}.

Taking the limit gives

\int_0^1x\,dx=\frac12.

Optional advanced note

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.

Interactive checku3a-riemann-limit-01

Evaluate $\lim_{n\to\infty}\frac{n+1}{2n}$ .

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Divide numerator and denominator by $n$ .

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**Visual study**
Rectangles converging to a fixed total. Show Riemann rectangles becoming narrower while the sum stabilizes.

After the explanation

Use the section idea

Reading lens

Treat a definite integral as the limit of structured approximations, with the sample rule and sign visible in every rectangle.

Mental model

Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.

Decision

Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.

Common trap

Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.

Check yourself

Can you construct the sum from a table or formula and predict whether it is high or low before calculating?

Source & rights

Original instruction with traceable references.

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