Integration & Accumulation

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 6. This unit is worth 17–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Riemann Sums & the Definite Integral14 min · 3 objectivesApproximate area under a curve with left, right, and midpoint Riemann sums · Interpret the definite integral as a limit of Riemann sums · Predict whether a sum overestimates or underestimates based on monotonicity
• Antiderivatives & the Fundamental Theorem15 min · 3 objectivesFind antiderivatives using the reverse power rule and basic formulas · Evaluate definite integrals with the Fundamental Theorem of Calculus · Include the constant of integration for indefinite integrals
• U-Substitution14 min · 3 objectivesRecognize an integrand as f(g(x))·g′(x), the reverse of the chain rule · Carry out u-substitution for indefinite and definite integrals · Change the limits of integration when substituting in a definite integral
• Accumulation Functions14 min · 3 objectivesInterpret an integral with a variable upper limit as an accumulation function · Differentiate an accumulation function with the Fundamental Theorem (Part 1) · Use accumulation to find a quantity from its rate of change

Formulas in Unit 6

Every term in Unit 6

All 17 terms we publish for Integration & Accumulation, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Riemann sums

Left, right, midpoint and trapezoidal approximations of area. Left sums underestimate an increasing function; right sums overestimate it.

u-substitution

Choose u so that du appears (up to a constant) in the integrand. For a definite integral, either change the limits or convert back before evaluating.

Trapezoidal rule

(Δx/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]. Overestimates where the function is concave up.

Definite integral as a limit of Riemann sums

∫f dx is the limit as the subinterval width goes to zero, which is why it represents exact accumulated change.

Fundamental Theorem of Calculus part 1

d/dx ∫ₐˣ f(t)dt = f(x). With a variable upper limit g(x), the chain rule gives f(g(x))·g′(x).

Fundamental Theorem of Calculus part 2

∫ₐᵇ f(x)dx = F(b) − F(a) for any antiderivative F. Connects accumulation to antidifferentiation.

Accumulation function

g(x) = ∫ₐˣ f(t)dt increases where f is positive and has a maximum where f changes from positive to negative.

Properties of definite integrals

Reversing limits negates the integral, an integral from a to a is zero, and integrals split at any interior point.

Common antiderivatives

∫xⁿdx = xⁿ⁺¹/(n+1) for n ≠ −1, ∫(1/x)dx = ln|x| + C, ∫eˣdx = eˣ + C. The absolute value in the log matters.

Net change theorem

∫ₐᵇ f′(x)dx = f(b) − f(a). The integral of a rate gives the total change in the quantity.

Interpreting an integral in context

"The total volume of water that entered the tank between t = 0 and t = 5 minutes was 40 liters." Include what accumulated, over what interval, with units.

Choosing u in a substitution

Pick the inner function whose derivative already appears, up to a constant multiple. If nothing cancels, the substitution is wrong.

Definite integral with substitution

Either convert the limits to u values or convert back to x before evaluating. Mixing limits and variables is a guaranteed error.

Splitting an integral at a sign change

For total distance or area between curves, split wherever the integrand changes sign, then add the absolute values.

Riemann sum from a table

The subintervals may have unequal widths. Multiply each function value by its own width rather than assuming a uniform Δx.

Over- or underestimate justification

Say why: a left sum underestimates because the function is increasing; the trapezoid rule overestimates because the graph is concave up.

Accumulation function analysis

g(x) = ∫ₐˣ f(t)dt has g′ = f, so g increases where f is positive and is concave up where f is increasing.

What examiners penalize here

• AP problems often give data in a table and ask for a specific Riemann sum. Read carefully whether left, right, or midpoint endpoints are required, and note that subintervals may have *unequal* widths — compute each rectangle’s width separately then.
• Verify any antiderivative by differentiating it back — if F′ returns the integrand, F is correct. This quick check catches reverse-power-rule slips (like forgetting to divide by the new exponent) before they cost points.
• On the exam, converting the limits to u-values is usually faster and less error-prone than back-substituting. Write the new limits explicitly next to the integral so you evaluate the u-antiderivative at the correct endpoints.
• For "how much is there at time b" problems, use final value = initial value + ∫[a to b] rate dt. Integrating the rate gives only the *change*; you must add the starting amount to get the total. This structure appears on nearly every AP integral free-response.

Practice Calculus AB

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Calculus AB

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Rules
3. Unit 3 · Composite & Implicit Differentiation
4. Unit 4 · Contextual Applications
5. Unit 5 · Analytical Applications
6. Unit 6 · Integration & Accumulation
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration