A right Riemann sum approximates a definite integral by splitting the interval into subintervals and using the function value at the right endpoint of each subinterval as the rectangle height; it overestimates increasing functions and underestimates decreasing ones (AP Calc Topic 6.2, LO 6.2.A).
A right Riemann sum is one of the four approximation methods named in the CED (LIM-5.A.2) for estimating a definite integral: left, right, midpoint, and trapezoidal. You chop the interval [a, b] into subintervals, build a rectangle on each one, and set each rectangle's height equal to the function's value at the right edge of that subinterval. Add up all the rectangle areas (height × width) and you have your approximation.
Here's the intuition that makes it click. For a function that's going uphill, the right edge is the tallest point of each subinterval, so every rectangle pokes above the curve and the sum overestimates the true area. For a function going downhill, the right edge is the shortest point, so the rectangles sit below the curve and the sum underestimates. The CED makes this exact idea testable (LIM-5.A.4): you have to be able to say whether your approximation is too big or too small based on the function's behavior. One more thing the CED is explicit about: subintervals don't have to be the same width. AP loves giving you a table with unevenly spaced x-values, so always compute each width separately instead of assuming a uniform Δx.
Right Riemann sums live in Topic 6.2 (Approximating Areas with Riemann Sums) in Unit 6, supporting learning objective 6.2.A: approximate a definite integral using geometric and numerical methods. This is the on-ramp to the entire concept of integration. Before you ever see an antiderivative, Riemann sums are how you make sense of what a definite integral actually is, which is accumulated area built from skinny rectangles. The skill matters beyond Topic 6.2 too, because functions on the exam often show up only as a table of values (LIM-5.A.1 says integrals can be approximated from graphical, numerical, analytical, or verbal representations). When all you have is a table, you can't integrate symbolically. A Riemann sum is the tool, and the right-endpoint version is one of the specific flavors the exam names and tests.
Keep studying AP Calculus Unit 6
Visual cheatsheet
view galleryLeft Riemann Sum (Unit 6)
The mirror twin. A left Riemann sum uses the left endpoint of each subinterval for the rectangle height, so its over/under behavior flips: left underestimates increasing functions and overestimates decreasing ones. Knowing both lets you sandwich the true integral between two bounds.
Mid-point Riemann Sum (Unit 6)
Same rectangle idea, but the height comes from the middle of each subinterval. Midpoint sums usually land closer to the true value than left or right sums because the rectangle's overshoot on one side roughly cancels the undershoot on the other.
Area Under Curve (Unit 6)
The right Riemann sum is a finite, chunky version of the area under a curve. Take the limit as the number of rectangles goes to infinity and the sum becomes the definite integral itself. That limit definition is how Topic 6.2 hands off to the rest of Unit 6.
Indefinite Integral (Unit 6)
These are two different answers to the same question. A Riemann sum gives a numerical estimate of a definite integral, while antiderivatives (via the Fundamental Theorem) give exact values. When a function is only given as a table, the Riemann sum is the only option.
Right Riemann sums are a recurring FRQ staple, almost always paired with a table of values. The 2021 FRQ Q1 (bacteria density in a petri dish), 2022 FRQ Q4 (melting ice sculpture), and 2023 FRQ Q1 (gas pumping rate) all asked for a right Riemann sum approximation using data from a table, often with nonuniform subinterval widths. The standard FRQ ask has two parts. First, compute the sum, showing the setup as (width)(right-endpoint value) for each subinterval. Second, interpret or judge it, like explaining what the approximation means in context with units, or stating whether it's an overestimate or underestimate and justifying with the function's behavior. Multiple-choice questions hit the same ideas more directly: which sum overestimates an increasing function (right), which underestimates a decreasing function (right), and how the rectangle heights are determined (the function value at each subinterval's right endpoint). Two grading traps to avoid: don't assume equal widths when the table spacing is uneven, and don't justify over/under estimates with concavity, which controls trapezoid and midpoint error, not left/right error. For left and right sums, it's increasing/decreasing that matters.
Both build rectangles on subintervals; the only difference is which endpoint sets the height. Right sums sample at each subinterval's right edge, left sums at the left edge. The consequence is what trips people up: for an increasing function, right overestimates and left underestimates, and the roles swap for a decreasing function. A quick sanity check is to sketch one rectangle. If the curve climbs into the rectangle's top-right corner, the right-endpoint rectangle sticks above the curve, so the right sum is too big.
A right Riemann sum approximates a definite integral by using the function's value at the right endpoint of each subinterval as the rectangle height.
For an increasing function, a right Riemann sum always overestimates the definite integral; for a decreasing function, it always underestimates it.
Subintervals can have unequal widths (LIM-5.A.2), so on table-based FRQs you must calculate each width from the given x-values instead of assuming a uniform Δx.
Justify over/under estimates for left and right sums using whether the function is increasing or decreasing, not concavity.
Released FRQs from 2021, 2022, and 2023 all asked for a right Riemann sum from a table, then asked for interpretation in context with correct units.
It's a method for approximating a definite integral (Topic 6.2). You divide the interval into subintervals and use the function's value at the right endpoint of each subinterval as the rectangle's height, then add up all the rectangle areas.
No. It overestimates only when the function is increasing on the interval. If the function is decreasing, a right Riemann sum underestimates the true value. The function's direction, not the sum type alone, determines over versus under.
A right sum takes rectangle heights from the right endpoint of each subinterval; a left sum takes them from the left endpoint. Their errors are opposites: for an increasing function, right overestimates while left underestimates.
For each subinterval, multiply its width (the difference between consecutive x-values in the table) by the function value at the right endpoint, then add the products. Watch for unequal spacing; FRQs like 2021 Q1 and 2023 Q1 use nonuniform tables on purpose.
No. Concavity determines error for trapezoidal and midpoint approximations. For right (and left) Riemann sums, whether the function is increasing or decreasing is what decides overestimate versus underestimate, and that's the justification graders want.