The trapezoidal rule approximates a definite integral by slicing the area under a curve into trapezoids instead of rectangles, then adding their areas. Each trapezoid's area is the average of its two heights times its width, which makes it the average of the left and right Riemann sums.
The trapezoidal rule is a way to estimate a definite integral when you can't (or don't want to) find an antiderivative. Instead of stacking rectangles under the curve like a Riemann sum does, you connect consecutive points on the curve with straight lines, creating trapezoids. Each trapezoid's area is the width of the subinterval times the average of the function values at its two endpoints, so the formula for one piece is (1/2)(f(a) + f(b))(b - a). Add up all the pieces and you have your estimate.
Here's the intuitive version. A left Riemann sum uses the left height, a right Riemann sum uses the right height, and the trapezoidal rule splits the difference by averaging them. Because the slanted top of each trapezoid hugs the curve more closely than a flat rectangle top, the trapezoidal rule is usually more accurate than a left or right sum with the same number of subintervals. On the AP exam you'll often apply it to a table of values, where the subintervals may not be equal width, so compute each trapezoid separately rather than reaching for a shortcut formula.
The trapezoidal rule lives in Unit 6 (Integration and Accumulation of Change), in the topic on approximating areas with Riemann sums, and it's tested on both AB and BC. It supports the core skill of estimating accumulated change from data, which is exactly what the famous AP table problems demand. When a problem gives you values of a rate function at scattered times (water flowing into a tank, a car's velocity, people entering a line), the trapezoidal rule lets you estimate the total accumulation without ever knowing the function's formula. It also feeds directly into interpretation skills, because the exam loves asking whether your estimate is an overestimate or underestimate, and answering that requires you to think about concavity. That's a quiet but powerful link back to derivative analysis from Units 4 and 5.
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Riemann Sum (Unit 6)
The trapezoidal sum is literally the average of the left and right Riemann sums. If you can compute those two, you can always recover the trapezoidal answer, which is a great way to check your work.
Definite Integral (Unit 6)
The trapezoidal rule approximates a definite integral, and as the number of subintervals grows, the estimate converges to the exact integral value. It's the bridge between discrete data and the continuous integral.
Total Distance Traveled (Unit 8)
When a table gives velocity values at certain times, a trapezoidal sum of speed estimates total distance traveled. This is one of the most common real exam setups for the rule.
Concavity and the Second Derivative (Units 4-5)
Whether a trapezoidal estimate is too big or too small depends on concavity. Concave up means the chords sit above the curve, so the trapezoids overestimate. Concave down means they underestimate.
The classic setup is a free-response table problem. You're given values of a function (often a rate) at a handful of input values, and the prompt says something like "use a trapezoidal sum with the subintervals indicated by the data in the table." Three things to nail. First, the subintervals are usually unequal, so compute each trapezoid as (1/2)(left value + right value)(width) rather than using a uniform-width shortcut. Second, include units and interpret your answer in context, since the interpretation point is often worth as much as the arithmetic. Third, be ready for the follow-up question asking whether your approximation overestimates or underestimates the true value, which you justify using concavity (or whether the rate function is increasing or decreasing, for rectangle sums). Multiple-choice questions test the same ideas, often asking you to compute a trapezoidal sum from a small table or graph, or to rank left, right, midpoint, and trapezoidal estimates against the true integral.
Riemann sums use rectangles whose heights come from one sample point per subinterval (left endpoint, right endpoint, or midpoint). The trapezoidal rule uses both endpoints of each subinterval and connects them with a slanted line, forming a trapezoid. The payoff is accuracy. The trapezoidal estimate equals the average of the left and right Riemann sums, so it usually lands closer to the true integral than either one alone. Technically the trapezoidal rule isn't a Riemann sum at all, since no single sample height generates a trapezoid, but the AP exam treats them as a family of approximation methods.
The trapezoidal rule estimates a definite integral by summing trapezoid areas, where each trapezoid's area is (1/2)(f(left) + f(right))(width).
It equals the average of the left and right Riemann sums, which makes it generally more accurate than either one.
On table-based FRQs the subintervals are often unequal widths, so calculate each trapezoid individually instead of using a one-size formula.
If the function is concave up on the interval, the trapezoidal rule overestimates the integral; if it's concave down, it underestimates.
When the function is a rate of change, the trapezoidal sum estimates total accumulated change, like total water in a tank or total distance traveled.
Always attach units and a contextual interpretation to your answer, because exam rubrics award points for both.
It's a method for approximating a definite integral by dividing the area under a curve into trapezoids and adding their areas. Each trapezoid contributes (1/2)(f(left) + f(right)) times the subinterval width, and it appears in Unit 6 alongside Riemann sums.
Not technically, since a Riemann sum uses one sample height per subinterval to build a rectangle, and no single height produces a trapezoid. But the trapezoidal sum equals the average of the left and right Riemann sums, so the AP exam groups them together as approximation methods.
Look at concavity. If the curve is concave up, the straight tops of the trapezoids sit above the curve, giving an overestimate. If it's concave down, the chords sit below the curve, giving an underestimate.
No, and on AP free-response table problems they usually aren't. Compute each trapezoid separately with its own width, like (1/2)(f(2) + f(5))(3) for a subinterval from x = 2 to x = 5, then add the pieces.
Yes. It's part of Unit 6 (Integration and Accumulation of Change) on both exams, most often tested through table-based free-response questions where you estimate an integral from data.