Difficulty: Intermediate | Prerequisites: basic understanding of definite integrals, summation notation, and function evaluation from a table.
Unit 6 sits at the transition from differential to integral calculus. Up to now the course has been about rates of change; here you learn to go the other way and accumulate quantities. The definite integral gives the net signed area between a curve and the x-axis, but you cannot always find an antiderivative, so you need numerical methods to approximate that area. Riemann sums and trapezoidal sums are the two families of approximation you will use, and the AP exam tests them regularly, both with formulas and with data tables.
When you cannot evaluate a definite integral exactly, you chop the interval into subintervals and add up the areas of simple shapes (rectangles or trapezoids) that approximate the curve. Left, right and midpoint Riemann sums use rectangles whose heights are sampled at different points; the trapezoidal sum averages the left and right endpoints to form trapezoids. All four methods appear on the AP exam, usually with a data table rather than a formula.
Definite integral
The net signed area between the graph of f(x) and the x-axis from x = a to x = b, written as the integral from a to b of f(x) dx. In simple terms, it tells you the total accumulated quantity over an interval.
Riemann sum
An approximation of a definite integral formed by dividing [a, b] into subintervals and summing the areas of rectangles built on those subintervals. Think of it as adding up the areas of tall, thin rectangles that roughly follow the shape of the curve.
Left Riemann sum (LRS)
A Riemann sum where the height of each rectangle is the function value at the left endpoint of each subinterval. In simple terms, you read the y-value from the left edge of each strip.
Right Riemann sum (RRS)
A Riemann sum where the height of each rectangle is the function value at the right endpoint of each subinterval. You read the y-value from the right edge of each strip.
Midpoint Riemann sum (MRS)
A Riemann sum where the height of each rectangle is the function value at the midpoint of each subinterval. You read the y-value from the centre of each strip, which often gives a better approximation than left or right.
Trapezoidal sum (trapezoidal rule)
An approximation that connects consecutive data points with straight lines, forming trapezoids instead of rectangles. Think of it as averaging the left and right heights for each strip, which tends to hug the curve more closely.
Subinterval
One of the smaller intervals you get when you partition [a, b] into n equal pieces. Each subinterval has width delta x = (b - a) / n.
Delta x (partition width)
The width of each subinterval: delta x = (b - a) / n. This is the base of every rectangle or trapezoid in the sum.
Identify the interval [a, b] and the number of subintervals n.
Compute delta x = (b - a) / n.
List the partition points: a, a + delta x, a + 2 delta x, ... , b.
When working from a table, the x-values are already given. Check that the spacing is equal (the problem will state "equal subintervals").
Use the function value at the left endpoint of each subinterval as the rectangle's height.
For n subintervals with endpoints x_0, x_1, ..., x_n, the left sum uses f(x_0), f(x_1), ..., f(x_{n-1}).
You drop the last data point.
Example from the homework table (h(x), x from -3 to 1, two subintervals): delta x = (1 - (-3)) / 2 = 2. Partition points: -3, -1, 1. Left sum = delta x * [h(-3) + h(-1)] = 2 * [5 + 2] = 14.
Use the function value at the right endpoint of each subinterval as the rectangle's height.
For n subintervals, the right sum uses f(x_1), f(x_2), ..., f(x_n).
You drop the first data point.
Example from the homework table (h(x), x from -3 to 9, three subintervals): delta x = (9 - (-3)) / 3 = 4. Partition points: -3, 1, 5, 9. Right sum = 4 * [h(1) + h(5) + h(9)] = 4 * [-3 + (-2) + 11] = 4 * 6 = 24.
Use the function value at the midpoint of each subinterval.
For this to work with a table, the midpoints must be among the given x-values. This constrains which subinterval counts are possible.
Example from the homework table (h(x), x from -3 to 9, three subintervals): delta x = 4. Subintervals: [-3, 1], [1, 5], [5, 9]. Midpoints: -1, 3, 7. Midpoint sum = 4 * [h(-1) + h(3) + h(7)] = 4 * [2 + (-7) + 6] = 4 * 1 = 4.
Instead of rectangles, connect consecutive points with a straight line and compute the area of each trapezoid.
Each trapezoid's area = (delta x / 2) * [f(x_i) + f(x_{i+1})].
Summing all trapezoids gives: (delta x / 2) * [f(x_0) + 2 f(x_1) + 2 f(x_2) + ... + 2 f(x_{n-1}) + f(x_n)].
The first and last values appear once; every interior value appears twice.
Example from the homework table (h(x), x from -3 to 3, three subintervals): delta x = (3 - (-3)) / 3 = 2. Trapezoidal sum = (2/2) * [h(-3) + 2 h(-1) + 2 h(1) + h(3)] = 1 * [5 + 4 + (-6) + (-7)] = -4.
When given a function like 2x sin(x), compute delta x from the interval, list the partition points, then evaluate the function at each required point (left endpoints, right endpoints, or midpoints).
Example: integral of 2x sin(x) from 0 to pi, four subintervals, right Riemann sum. Delta x = pi/4. Right endpoints: pi/4, pi/2, 3pi/4, pi. Evaluate 2x sin(x) at each, multiply each by pi/4, and add.
Some tables have unequal spacing (e.g. x = -2, 0, 1, 3, 5, 8, 9). The problem will specify how many subintervals to use, and you choose the grouping that makes them equal.
For three subintervals over [-2, 9] with midpoint sum, look for groups whose midpoints appear in the table. This is common on the AP free-response section.
All formulas assume n equal subintervals of width delta x = (b - a) / n, with partition points x_0 = a, x_1, x_2, ..., x_n = b.
\text{Left Riemann Sum} = \Delta x \bigl[f(x_0) + f(x_1) + \cdots + f(x_{n-1})\bigr]\text{Right Riemann Sum} = \Delta x \bigl[f(x_1) + f(x_2) + \cdots + f(x_n)\bigr]\text{Midpoint Riemann Sum} = \Delta x \Bigl[f\!\Bigl(\frac{x_0+x_1}{2}\Bigr) + f\!\Bigl(\frac{x_1+x_2}{2}\Bigr) + \cdots + f\!\Bigl(\frac{x_{n-1}+x_n}{2}\Bigr)\Bigr]\text{Trapezoidal Sum} = \frac{\Delta x}{2}\bigl[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\bigr]Quick memory trick: the trapezoidal sum is the average of the left and right Riemann sums. If you have already computed both, just add them and divide by two.
Engineers and scientists use these same approximation methods whenever they have discrete sensor readings (temperature every 5 seconds, velocity every 0.1 km) and need a total quantity (total heat, total distance). The trapezoidal rule is the workhorse of numerical integration in software libraries because it is simple, fast, and accurate enough for most practical measurements.
Students often think the trapezoidal sum requires dividing by 2 at the very end. It does not. The factor of 1/2 is baked into the formula (delta x / 2), and the interior values are doubled to compensate. If you divide the final answer by 2 as well, you halve the correct result.
Students sometimes include all data points for a left or right sum. A left sum drops the last point; a right sum drops the first. Mixing this up adds an extra term and gives the wrong answer.
When using a midpoint sum from a table, students sometimes pick the midpoint of the x-values listed rather than the midpoint of the subinterval. The midpoint must be calculated from the subinterval boundaries.
Students confuse the number of subintervals with the number of data points. If you have 7 data points equally spaced, you have 6 subintervals, not 7.
Riemann sums and trapezoidal approximations appear on nearly every AP Calculus AB exam, both multiple choice and free response.
Free-response questions typically give a table of values and ask for one specific approximation method. You must show the setup (the formula with numbers substituted) to earn full marks.
Know how to determine whether an approximation is an overestimate or underestimate. For a function that is increasing, a left Riemann sum underestimates and a right Riemann sum overestimates. For a function that is concave up, the trapezoidal sum overestimates.
The exam may ask you to use unequal subintervals. In that case, delta x changes per subinterval and you cannot factor it out.
True or false: A left Riemann sum uses the function value at the right endpoint of each subinterval. (False.)
Fill in the blank: The trapezoidal sum is equal to the average of the ______ and ______ Riemann sums. (Left, right.)
True or false: If a function is decreasing on [a, b], the left Riemann sum overestimates the integral. (True.)
Fill in the blank: For n equal subintervals on [a, b], delta x = ______. ((b - a) / n.)
True or false: In the trapezoidal formula, every function value is doubled. (False. The first and last values appear once; only the interior values are doubled.)
This material connects directly to the Fundamental Theorem of Calculus (Unit 6, later sections), which gives you the exact value of the integral when an antiderivative exists. Riemann sums are the conceptual foundation: the definite integral is defined as the limit of a Riemann sum as n approaches infinity. Accumulation functions and average value of a function both build on the definite integral concept introduced here.
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