---
title: "Trapezoidal Sum — AP Calculus Definition & Exam Guide"
description: "A trapezoidal sum approximates a definite integral using trapezoids instead of rectangles. Learn the formula, the concavity rule, and how AP Calc FRQs test it."
canonical: "https://fiveable.me/ap-calc/key-terms/trapezoidal-sum"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 6"
---

# Trapezoidal Sum — AP Calculus Definition & Exam Guide

## Definition

A trapezoidal sum approximates a definite integral by replacing rectangles with trapezoids on each subinterval, averaging the left and right function values and multiplying by the width. On the AP exam (Topic 6.2), it's usually computed from a table, often with unequal subinterval widths.

## What It Is

A trapezoidal sum is one of the four approximation methods the CED names for estimating a [definite integral](/ap-calc/unit-6/approximating-areas-with-riemann-sums/study-guide/juN9YbvFYlJtpsMl "fv-autolink") (LIM-5.A.2), alongside left, right, and midpoint Riemann [sums](/ap-calc/unit-1/determining-limits-using-algebraic-properties-limits/study-guide/HjStgVKViPGZj1CxYwEB "fv-autolink"). Instead of capping each subinterval with a flat rectangle top, you connect the two endpoints of the function with a straight line, which turns each piece into a trapezoid. The area of each trapezoid is the average of the two function values times the width: (1/2)[f(left) + f(right)] · Δx. Add up all the trapezoids and you have your estimate.

Here's the intuitive version: a trapezoidal sum is just the average of the [left Riemann sum](/ap-calc/key-terms/left-riemann-sum "fv-autolink") and the right Riemann sum (when the partition is uniform). The left sum errs one way, the right sum errs the other, and the trapezoid splits the difference. That's why it's usually the most accurate of the four methods. One more thing the CED makes explicit (LIM-5.A.2): partitions don't have to be uniform. AP tables love giving you subintervals of different widths, so never assume Δx is constant. Read it off the table for each piece.

## Why It Matters

Trapezoidal sums live in Topic 6.2 (Approximating Areas with Riemann Sums) in [Unit 6](/ap-calc/unit-6 "fv-autolink"): [Integration and Accumulation of Change](/ap-calc/unit-6/integration-accumulation-change/study-guide/NWRV9MaRJO4Eno32l5Xp "fv-autolink"), supporting learning objective 6.2.A. The big idea behind LIM-5.A.1 is that definite integrals show up for functions you can't always write a formula for. If all you have is a table of values (a tree's height, a particle's velocity, a student's reading rate), you can't antidifferentiate anything. Approximation is your only move, and the trapezoidal sum is the College Board's favorite version of it. It also connects to LIM-5.A.4, because the function's concavity tells you whether your trapezoidal estimate is an overestimate or underestimate, which the exam asks about constantly.

## Connections

### Left and Right Riemann Sums (Unit 6)

A trapezoidal sum on a uniform partition is literally the average of the left and right Riemann sums. If you've already computed both, you can get the trapezoidal answer for free by averaging them.

### Concavity and the Second Derivative (Unit 5)

The over/[underestimate](/ap-calc/unit-4/approximating-values-function-using-local-linearity-linearization/study-guide/mNrv8hwdyTqGWapbyqAH "fv-autolink") question is a concavity question in disguise. The trapezoid's slanted top sits above a concave-up curve (overestimate) and below a concave-down curve (underestimate). Increasing vs. decreasing controls left and right sums; concavity controls trapezoids.

### Accumulation Functions and Definite Integrals (Unit 6)

Trapezoidal sums approximate the same thing the Fundamental Theorem computes exactly. On FRQs, the integral you're estimating usually means something physical, like total meters traveled or total pages read, so you need to interpret the number with units, not just calculate it.

### Average Value and Rate-In/Rate-Out Models (Unit 8)

Table-based FRQs often pair a trapezoidal sum part with an average value or total [accumulation](/ap-calc/unit-8/using-accumulation-functions-definite-integrals-applied-contexts/study-guide/nUlJKvXqRcsfLnVMd5fG "fv-autolink") part later in the same problem. The trapezoidal estimate of ∫R(t)dt is your estimate of the total change in the quantity R measures.

## On the AP Exam

This is a workhorse FRQ skill. The 2018 FRQ (tree height table), 2019 FRQ (particle velocity table), and 2025 FRQ on both AB and BC (reading rate) all asked for a trapezoidal sum from a table of selected values. The pattern is consistent: you're given 4-5 values of a function, often with nonuniform spacing, and asked to use a trapezoidal sum to approximate a definite integral, then interpret the result in context with units. Show the setup, not just the answer; graders award points for the sum written out as (1/2)(f(a)+f(b))(b−a) terms. Multiple-choice questions hit the conceptual side: which method overestimates an increasing function (right sum, not trapezoid), how a trapezoid's area is computed, and why the trapezoidal sum tends to beat left, right, and midpoint sums for accuracy. Expect an over/underestimate justification too, and remember the justification for a trapezoidal sum must mention concavity, not increasing/decreasing.

## trapezoidal sum vs Midpoint Riemann sum

Both are 'the more accurate ones,' so they blur together. A midpoint sum uses one sample point (the middle of each subinterval) and still builds rectangles. A trapezoidal sum uses both endpoints and builds trapezoids. Also, the trapezoidal sum is the average of the left and right sums, not the midpoint sum. The midpoint sum is a genuinely different calculation, and on a table problem you often can't even do a midpoint sum because the midpoint values aren't given. That's exactly why FRQ tables ask for trapezoidal sums.

## Key Takeaways

- Each trapezoid's area is the average of the two endpoint function values times the subinterval width: (1/2)[f(left) + f(right)] · Δx.
- On a uniform partition, the trapezoidal sum equals the average of the left and right Riemann sums.
- Concavity decides the error direction: a trapezoidal sum overestimates a concave-up function and underestimates a concave-down function.
- AP tables frequently use nonuniform partitions, so compute each subinterval's width separately instead of assuming a constant Δx.
- Table-based FRQs (like 2018, 2019, and 2025) expect you to show the full trapezoidal sum setup and interpret the answer with correct units.
- Increasing or decreasing tells you about left and right sum errors, but it tells you nothing about trapezoidal sum error. Only concavity does.

## FAQs

### What is a trapezoidal sum in AP Calculus?

It's a way to approximate a definite integral by topping each [subinterval](/ap-calc/unit-6/riemann-sums-summation-notation-definite-integral-notation/study-guide/RnM03H2k6l3ewvxX "fv-autolink") with a slanted line instead of a flat rectangle, giving trapezoids with area (1/2)[f(left) + f(right)] · width. It's listed in the CED (LIM-5.A.2) alongside left, right, and midpoint Riemann sums.

### Is a trapezoidal sum always more accurate than a Riemann sum?

Usually but not always. Because it averages the left and right sums, it typically beats both, and AP questions frame it as the closest of the four methods. For some functions a midpoint sum can be just as good or better, but for AP purposes, trapezoidal is the accuracy answer.

### Does a trapezoidal sum overestimate or underestimate?

It depends on concavity, not on whether the function is increasing. Concave up means the trapezoid tops sit above the curve, so you overestimate. Concave down means you underestimate. Increasing/decreasing is the rule for left and right sums, a totally separate test.

### How is a trapezoidal sum different from a midpoint Riemann sum?

A midpoint sum builds rectangles using the function value at each subinterval's center; a trapezoidal sum uses both endpoints and connects them with a line. On table FRQs you usually only get endpoint values, which is why those problems ask for trapezoidal sums.

### Do AP Calc FRQs actually ask for trapezoidal sums?

Yes, regularly. The 2018, 2019, and 2025 exams all had FRQs requiring a trapezoidal sum from a table of values (tree height, particle velocity, and reading rate), often with unequal subinterval widths and an interpretation-with-units follow-up.

## Related Study Guides

- [6.2 Approximating Areas with Riemann Sums](/ap-calc/unit-6/approximating-areas-with-riemann-sums/study-guide/juN9YbvFYlJtpsMl)

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