Approximating Areas with Riemann Sums AB & BC
A Riemann sum replaces the region under $f$ with $n$ rectangles. With equal widths, $\Delta x = \frac{b-a}{n}$, the grid holds $n + 1$ points and the sum holds $n$ terms, and the sum is $\Delta x$ times the total of the sampled heights. The left sum samples every grid point but the last, the right sum every grid point but the first, and the midpoint sum samples points that are not on the grid at all. The trapezoidal rule averages the left and right sums, which is the same as $\frac{\Delta x}{2}$ times $f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)$.
The direction of the error is a fact about the function, not about the name of the sum. On an increasing $f$ the left sum underestimates and the right sum overestimates, and on a decreasing $f$ both verdicts swap. Trapezoids are decided by a different feature entirely: the trapezoid's top is a chord, so it overestimates on a concave-up $f$ and underestimates on a concave-down one, with the midpoint rule leaning the other way and by roughly half as much. If the function changes direction, or its concavity changes, inside the interval, there is no verdict to give.
The work
3 ways in · any order
Lesson
Approximating Areas with Riemann Sums
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Builds the subinterval width and the three standard sample choices, counts grid points against subintervals, then separates the two direction rules that get confused: monotonicity decides left and right rectangles, concavity decides trapezoids. Closes with data tables, where there is no single width at all.
Diagnostic
10-item topic check
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Ten items spanning the three failure modes of this topic: setup errors in the width, the sample points or the term count, deciding a rectangle sum's direction from habit rather than from whether the function increases, and assuming the trapezoidal rule always leans one way.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.