Calculus AB — approximating integrals from tables and accumulation functions

Two kinds of question from Unit 6 of AP Calculus AB appear on nearly every exam. In one, a table gives a few measured values of a rate and you approximate the total with a Riemann or trapezoidal sum, usually on subintervals of different widths. In the other, a function is defined as g(x), the integral of f from a fixed number to x, and f is given only as a drawn graph; the questions ask for values of g, its derivatives, its extrema. This material trains both, following topics 6.1 to 6.7 of the course framework.

The first half of the twelve quiz questions works from data. A tank collects rainwater at a measured rate at hours 0, 3, 5, 9 and 10, and you compute the left sum, the right sum and the trapezoidal sum on those four unequal subintervals; another function, known at five equally spaced times, gives a midpoint sum with two subintervals. Two questions ask which approximation must be too large or too small, one for an increasing and concave-down function and one for a decreasing function, and one asks what the integral of a rate in liters per minute measures.

The second half uses one drawn graph of f made of straight segments, crossing the axis at t = −1 and t = 3. From it you find g(4) as a sum of signed areas, g(−1) where the upper limit lies below the lower one and the sign flips, g′(5) and g″(5) as the height and the slope of the graph of f, the absolute maximum of g on the whole interval by the candidates test, and the derivative of an integral whose upper limit is x², where the chain rule adds a factor.

Wrong options are built from the slips these questions cause: adding table values without multiplying by the widths, treating unequal widths as equal, forgetting to halve in a trapezoid, taking the endpoints instead of the midpoints, counting area below the axis as positive, ignoring the reversed limits, reporting a value of f where g was asked for, forgetting the chain-rule factor. Each explanation writes the sum or the areas out in full.

The flashcards give the formula for each of the four sums on unequal subintervals, the Fundamental Theorem for g and its chain-rule form, and the six over- and underestimate rules for increasing, decreasing, concave up and concave down functions.

This is independent practice written by Zestly, based on the published AP Calculus AB course framework; it is not produced or endorsed by the College Board.

• Compute left, right, midpoint and trapezoidal sums from a table, including unequal subintervals
• Decide whether an approximation over- or underestimates from monotonicity and concavity
• Interpret the integral of a rate as a net change with units
• Evaluate an accumulation function from signed areas under a drawn graph, including reversed limits
• Find g′ and g″ of an accumulation function from the graph of f
• Locate the absolute maximum of an accumulation function
• Differentiate an integral with a variable upper limit using the chain rule

Practice material written by Zestly, based on the College Board AP Calculus AB course framework (Unit 6: Integration and Accumulation of Change, topics 6.1–6.7: accumulation, Riemann and trapezoidal sums, accumulation functions and the Fundamental Theorem of Calculus), 2026–27 course and exam description.