Writing a Riemann sum for a definite integral connects geometric accumulation with algebraic structure, allowing students to express integrals as limits based on partitions and sample-point products.

Writing a Riemann Sum for a Given Integral

This subsubtopic centers on translating the notation of a definite integral into the structure of a Riemann sum, which expresses accumulated area as the limit of products of function values and subinterval widths. AP Calculus AB students must understand how to identify the interval, construct partitions, choose sample points, and form the summation that represents the integral’s limiting behavior.

Understanding the Structure of a Definite Integral

A definite integral $\int_a^b f(x),dx$ represents accumulated change or net area over the interval $[a, b]$. To rewrite this integral as a Riemann sum, each component of the integral must correspond to a feature of the sum: the endpoints become the domain of partitioning, the integrand becomes the sampled function, and the differential $dx$ becomes a subinterval width.
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Partition of the interval [a,b][a,b][a,b] into subintervals [x0,x1],[x1,x2],…,[x4,x5] [x_0,x_1], [x_1,x_2], \dots, [x_4,x_5][x0​,x1​],[x1​,x2​],…,[x4​,x5​] under a curve y=f(x)y = f(x)y=f(x). The shaded strips illustrate the subdivision of the interval before forming a Riemann sum. This image focuses on the partition itself rather than the rectangle heights used in subsequent steps. Source.

Key Components Required for Translation

To express an integral as a Riemann sum, identify the following elements:

• Interval of integration: The closed interval $[a, b]$ over which accumulation occurs.

• Partition of the interval: A division of $[a, b]$ into $n$ subintervals, each with width $\Delta x$.

• Sample points: Chosen points within each subinterval, typically left endpoints, right endpoints, or midpoints, though any interior point is valid.

• Summand expression: A product that approximates area, formed as $f(\text{sample point})\cdot \Delta x$.

• Limit as $n \to \infty$: Ensures the approximation becomes exact.

These elements allow students to match the symbolic integral to its corresponding limit-based form.

Partitioning the Interval

To prepare a Riemann sum, divide the interval $[a, b]$ into $n$ subintervals. The width of each subinterval is constant for uniform partitions and is defined using the formula in the equation block below.

This width determines how the sample points are located and dictates the spacing of function evaluations within the Riemann sum.

A non-equation sentence follows naturally here, emphasizing that defining $\Delta x$ is essential because it directly links the original integral to each term in the summation.

Choosing Sample Points

Selecting sample points allows the Riemann sum to mirror the behavior of the integrand. If $x_i$ denotes the right endpoint of the $i$-th subinterval, then $x_i = a + i\Delta x$.

A sample point x1\*x_1^\*x1\*​ chosen within the first subinterval determines the height f(x1\*)f(x_1^\*)f(x1\*​) of the corresponding rectangle. Dashed lines mark the connection between the sample point on the x-axis and the function value on the curve. This emphasizes how one term of a Riemann sum, f(xi\*)Δxif(x_i^\*)\Delta x_if(xi\*​)Δxi​, is constructed. Source.

If using left endpoints, the sample point is $x_{i-1} = a + (i-1)\Delta x$, and if using midpoints, the point is $x_{i-1/2} = a + \left(i - \tfrac12\right)\Delta x$. While the choice of sample point affects approximation accuracy, all lead to the same limit as $n$ grows large.

Building the Summand

Each term in a Riemann sum represents a thin rectangle whose height is determined by the function value at the chosen sample point and whose width is $\Delta x$. The summand always follows the structure:

• Function value: $f(\text{sample point})$

• Subinterval width: $\Delta x$

• Product: $f(\text{sample point})\Delta x$

This product approximates the area contributed by one subinterval. Summing over all subintervals accumulates the contributions across the entire interval.

Constructing the Full Riemann Sum

Combine the elements discussed above to create the general form of the Riemann sum corresponding to the integral. A Riemann sum is built using summation notation that aggregates each small product of height and width across $n$ subintervals. The structure is expressed using the formula in the equation block below.

This summation provides the algebraic expression required before taking the limit, reflecting how area is approximated via many narrow rectangles.

A left Riemann sum for y=x3y = x^3y=x3 on [0,2][0,2][0,2] using four equal-width rectangles. Each rectangle height is set by the function value at the left endpoint of the subinterval, illustrating the general summation form f(xi)Δxf(x_i)\Delta xf(xi​)Δx. The explicit function y=x3y = x^3y=x3 adds detail beyond the syllabus but directly supports the concept of constructing Riemann sums. Source.

Writing a single integral as a Riemann sum requires clear alignment between the integral’s bounds, the partition’s structure, and the function evaluations.

Taking the Limit to Match the Integral

Once the summation is established, the limit as $n\to\infty$ completes the translation from approximation to exact accumulation. The limit removes the dependence on the number of subintervals and ensures the sum converges to the integral’s value. The complete limit expression mirrors the definite integral both conceptually and structurally.

Practical Steps for Students

When asked to convert a definite integral to a Riemann sum, proceed with the following structured approach:

• Identify the integral’s bounds $a$ and $b$.

• Determine the subinterval width using $\Delta x = \frac{b - a}{n}$.

• Choose a consistent sample point formula such as $x_i = a + i\Delta x$.

• Write the summand as $f(x_i)\Delta x$.

• Express the integral as the limit of the sum as $n\to\infty$.