Riemann Sums

The central idea

Partition $[a,b]$ by $a=x_0<x_1<\cdots<x_n=b$. Choose a sample point $x_i^\in[x_{i-1},x_i]$ and form $\sum_{i=1}^n f(x_i^)\Delta x_i$. If these sums approach one common value as the mesh tends to zero, $f$ is Riemann integrable and that value is $\int_a^b f(x)dx$.

Definitions, hypotheses, and notation

Sample-point notation separates partition geometry from function evaluation. The width $\Delta x_i$ belongs to the subinterval, while $f(x_i^*)$ is the sampled height. On a nonuniform partition the widths differ, so replacing all of them by $(b-a)/n$ changes the sum. Integrability asserts that sufficiently fine choices converge to one value.

Error direction can be predicted for monotone functions: left and right sums bracket the integral when all subintervals have equal orientation. Concavity helps compare midpoint and trapezoidal approximations. These predictions are useful verification tools, but the finite sum is still an approximation unless a limit is taken or a special exactness property applies.

Conceptual meaning

Each term is signed rectangle area: height times width. Refining the partition reduces the geometric mismatch. Left, right, and midpoint rules differ in sample point but converge to the same integral for continuous functions.

A dependable method and decision rule

1. Compute $\Delta x=(b-a)/n$ for an equal partition.
2. Write the sample point: left $a+(i-1)\Delta x$, right $a+i\Delta x$, or midpoint $a+(i-1/2)\Delta x$.
3. Substitute the sample point into $f$.
4. Include the width factor and sum from $i=1$ through $n$.
5. Simplify finite sums or take the limit as $n\to\infty$ when requested.

Fully worked example

Graphical or geometric meaning

For an increasing function, left rectangles sit mostly below the graph and right rectangles mostly above it. Midpoint rectangles balance more of the local error. Negative function values produce signed rectangles below the axis.

Common mistakes and why they fail

Verification and reasonableness checks

• Count exactly $n$ subintervals and $n$ sample points.
• Verify widths add to $b-a$.
• Use monotonicity to predict whether a left or right sum is high or low.

Indexing determines sample points

For a uniform partition, $\Delta x=(b-a)/n$. Right endpoints are $a+i\Delta x$ for $i=1,\ldots,n$; left endpoints use $i=0,\ldots,n-1$; midpoints are $a+(i-\tfrac12)\Delta x$. Writing these before evaluating $f$ prevents off-by-one errors. If $f$ is increasing and nonnegative, left sums underestimate and right sums overestimate, providing a qualitative check. A definite integral appears only after the maximum subinterval width tends to zero. Retain the factor $\Delta x$ in every finite sum and describe the result as an approximation unless exact equality has been justified separately.

Practice

1. For $[1,3]$ with $n=4$, find $\Delta x$.
2. Write the right endpoint $x_i$ on $[0,1]$ with $n$ equal pieces.
3. For increasing $f$, which is larger: left or right sum?

Connections and next steps

• Sigma Notation
• Definite Integral
• Area Under a Curve