How to Calculate a Riemann Sum

You have a function, an interval, and a number of rectangles n. Your exam expects you to produce a number, and here is how to calculate a Riemann sum by hand, with the arithmetic laid out so you do not lose a term.

The formula is Σ_{i=1}^{n} f(x_i*) · Δx, where Δx = (b - a)/n and x_i* is a sample point inside the i-th subinterval. The process is built on four steps, each a simple check on your scratch paper.

This is the width of every rectangle. For a uniform partition, it is a single number. For the interval [0,2] with n = 4, Δx = (2-0)/4 = 0.5. For [0,π] with n = 4, Δx = (π - 0)/4 ≈ 0.7854. Write it down before you do anything else. If you miscompute Δx, every subsequent term is wrong.

The failure case: on an exam, a student writes Δx = (b - a) without dividing by n. This single error multiplies every rectangle height by n, giving a sum that is off by a factor of 4 or 6 or 10. Check your denominator.

Start at x₀ = a. Add Δx repeatedly until you reach xₙ = b. For [0,2] with Δx = 0.5, the list is 0, 0.5, 1.0, 1.5, 2.0. There are n + 1 points: five points for n = 4. The most common indexing mistake is using n points instead of n + 1, which drops the last endpoint and shifts every subinterval.

Write the list vertically down the margin of your scratch paper. You will refer to it for every sample point decision.

The type of Riemann sum determines which x_i* you use from each subinterval [x_{i-1}, x_i].

Left Riemann Sum

Use x_i* = x_{i-1} (the left endpoint). For n = 4, you use x₀, x₁, x₂, x₃, four points, skipping xₙ. For an increasing function, a left Riemann sum underestimates the area; for a decreasing function, it overestimates.

Right Riemann Sum

Use x_i* = x_i (the right endpoint). You use x₁, x₂, x₃, x₄, again four points, skipping x₀. For an increasing function, a right Riemann sum overestimates the area; for a decreasing function, it underestimates.

Midpoint Riemann Sum

Use the midpoint of each subinterval: (x_{i-1} + x_i)/2. You evaluate the function at four different points, none of which are the original partition points. The midpoint rule is often more accurate than the left or right rule for the same n, but its over/underestimate behavior depends on concavity, not monotonicity.

For each sample point, compute f(x_i*). Multiply each by Δx. Add them all. That single number is your Riemann sum.

Set it up as a table so you can see every term. A common hand-computation error is to sum the function values and then multiply by Δx only once, that gives you n times the correct answer if you multiply at the end, or no factor at all if you forget entirely. Every individual term is f(x_i*) · Δx. Write each product in its own column.

OpenStax Calculus Volume 1, section 5.1, shows this exact function to introduce left, right, and midpoint approximations.

Given: f(x) = x², [a,b] = [0,2], n = 4. Δx = 0.5. Partition points: 0, 0.5, 1.0, 1.5, 2.0.

Left Riemann Sum

Sample points: 0, 0.5, 1.0, 1.5. Function values: f(0) = 0, f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25. Multiply each by Δx = 0.5: 0, 0.125, 0.5, 1.125. Sum = 1.75. The exact integral is 8/3 ≈ 2.667, so the left sum is an underestimate. Because f is increasing on [0,2], the left Riemann sum underestimates the area.

Right Riemann Sum

Sample points: 0.5, 1.0, 1.5, 2.0. Function values: f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25, f(2) = 4. Multiply by Δx = 0.5: 0.125, 0.5, 1.125, 2.0. Sum = 3.75. This is an overestimate. For an increasing function, the right Riemann sum always overestimates the area.

The table format keeps the arithmetic clean. Write each subinterval as a row: i, [x_{i-1}, x_i], x_i*, f(x_i*), product. The AP Calculus AB released free-response questions use this layout in their scoring guidelines, setup points are awarded before any arithmetic.

This example tests trigonometric evaluation and midpoint selection. Δx = (π - 0)/4 = π/4 ≈ 0.7854. Partition points: 0, π/4, π/2, 3π/4, π.

Midpoints: x₁* = π/8 ≈ 0.3927, x₂* = 3π/8 ≈ 1.1781, x₃* = 5π/8 ≈ 1.9635, x₄* = 7π/8 ≈ 2.7489. Function values: sin(0.3927) ≈ 0.3827, sin(1.1781) ≈ 0.9239, sin(1.9635) ≈ 0.9239, sin(2.7489) ≈ 0.3827. Multiply each by Δx ≈ 0.7854: 0.3006, 0.7256, 0.7256, 0.3006. Sum ≈ 2.052.

The exact integral of sin(x) from 0 to π is 2.052. Compare that to the left sum (error about 0.667) or right sum (error about 1.333) for the same n. The midpoint rule is substantially more accurate for smooth functions. Because sin(x) on [0,π] is concave down on (0,π/2) and concave up on (π/2,π), the midpoint rule does not guarantee a simple over/underestimate, check the concavity of your function before making a justification.

Use a scientific calculator in radian mode. The most frequent failure in trig Riemann sums is degrees-versus-radians: if your calculator is in degrees, sin(0.3927) is about 0.007, not 0.3827, and the sum collapses to near zero.

Draw a table with columns: i (from 1 to n), subinterval [x_{i-1}, x_i], sample point x_i*, f(x_i*), and product f(x_i*)·Δx. This forces you to write every term explicitly. The table also catches indexing errors: for a left sum, the sample point column should never contain xₙ; for a right sum, it should never contain x₀.

On an AP exam, the scoring rubric for a table-based Riemann sum awards points for the correct identification of each subinterval's endpoints and the correct sample point selection, even if the arithmetic is wrong. The table proves you know the setup.

n vs. n + 1 points: A partition has n + 1 endpoints. For a left sum, you use the first n endpoints. For a right sum, you use the last n endpoints. Using n points when you need n + 1 (or vice versa) shifts every subinterval.

Wrong endpoints: Using x_i when the problem asks for left endpoints, or x_{i-1} when it asks for right endpoints. This is the most heavily penalized error on free-response questions, it is a concept error, not an arithmetic slip.

Forgetting the Δx factor: Summing function values without multiplying by Δx. The result has the wrong dimension and is off by a factor equal to the subinterval width. A student who writes "sum = 3.5" without multiplying by 0.5 would report 3.5 instead of 1.75 for the left sum of x² on [0,2].

Sign errors: A Riemann sum computes signed area. If f(x) is negative on any subinterval, the product f(x_i*)·Δx is negative. The total sum can be smaller than the positive area or even negative. The definite integral is not always the area under the curve, it is the net signed area.

Radians vs. degrees: For any trigonometric function, set your calculator to radian mode. Using degrees changes every function value and produces a meaningless approximation.

Over/underestimate justification by monotonicity, not concavity: For left and right sums, the over/underestimate behavior depends on whether the function is increasing or decreasing on the entire interval. For midpoint sums, it depends on concavity. The AP Calculus CED Topic 6.2 specifies: "If f is increasing on [a,b], then left Riemann sum is an underestimate and right Riemann sum is an overestimate." Do not mention concavity unless the problem asks about the midpoint or trapezoidal rule.

After you finish by hand, verify your result with a Riemann sum calculator. Enter the function, the interval, the number of subintervals, and the rule (left, right, or midpoint). The calculator returns the sum instantly. If your hand-computed sum differs by more than 0.01 (for n = 4), you have an arithmetic error or a misidentified sample point. Recheck each row of your table.