Topic 6.2 Approximating Areas with Riemann Sums: approximate area using left, right, midpoint, and trapezoidal sums, and reason about over- and under-estimates.

What this topic is asking

The College Board (Topic 6.2) introduces Riemann sums: approximating the area under a curve by rectangles or trapezoids. You must compute left, right, midpoint, and trapezoidal approximations, often from a table, and reason about whether each over- or under-estimates the true area.

The four approximations

A worked Riemann sum

Over- and under-estimation

Why unequal widths and table problems dominate

On the AP exam, Riemann sums almost always come from a table of values, and the subintervals are usually of unequal width. This is a deliberate test of whether you multiply each height by the correct width rather than assuming a uniform $\Delta x$. The single most common error is using one width for all terms. Read the partition points from the table, compute each width as the difference of consecutive $x$-values, and pair each width with the correct endpoint height. The trapezoidal sum is then just the average of the left and right sums when the same partition is used, which is a useful check.

The midpoint sum and its accuracy

The midpoint sum uses the function value at the center of each subinterval as the rectangle height. It tends to be more accurate than left or right sums because the rectangle's overshoot on one side of the curve roughly cancels its undershoot on the other. On a table problem the midpoint sum is only usable when the table actually provides the values at the subinterval midpoints; you cannot compute it from endpoint data alone. When the problem supplies a formula rather than a table, the midpoint sum is straightforward to evaluate. Knowing when each method is even possible, given the data, is part of selecting the right approximation under exam conditions.

Interpreting the approximation in context

Riemann sums frequently model a real accumulation, such as estimating total water from a table of flow rates or total distance from sampled velocities. In that setting the sum approximates the definite integral of the rate, so its units are rate times time, and the result should be reported with units and a contextual sentence. The over/under reasoning then has a physical meaning too: a left sum of an increasing flow rate underestimates the true volume accumulated. Connecting the numerical estimate back to the quantity it approximates, and stating whether your estimate is likely high or low, is exactly the kind of interpretation the free-response section rewards.