End of Unit 6 CA Integration and Accumulation of Change Guide

The end of Unit 6 CA Integration and Accumulation of Change marks the close of AP Calculus AB’s longest unit and the point where differentiation and integration finally connect. If you’re preparing for this checkpoint, here’s the direct answer: you need to command the Fundamental Theorem of Calculus, definite and indefinite integrals, Riemann sums, the trapezoidal rule, accumulation functions, average value, and u-substitution. These topics form the backbone of a substantial share of the AP Calculus AB exam, so mastery here pays off twice.

What Unit 6 Actually Covers

Unit 6 in AP Calculus AB is titled “Integration and Accumulation of Change.” According to the College Board course framework, it spans from approximating areas with Riemann sums all the way to applying integration to motion and accumulation problems. It’s the unit where you stop treating integration as “reverse differentiation” and start treating it as a tool for measuring net change.

The unit divides into roughly six conceptual chunks:

• Approximating integrals with Riemann sums (left, right, midpoint, trapezoidal)
• Defining the definite integral as a limit of Riemann sums
• The Fundamental Theorem of Calculus (both parts)
• Antiderivatives and indefinite integrals
• Accumulation functions and net change
• Integration by substitution (u-substitution)

Each builds on the last, so gaps early in the unit tend to resurface on the end-of-unit assessment.

Riemann Sums and the Definite Integral

A Riemann sum estimates the area under a curve by slicing the interval into subintervals and adding up rectangles. The width of each subinterval is Δx = (b − a)/n, and the height comes from evaluating the function at a chosen point in each subinterval.

The four common variants differ only in where you sample the height:

• Left Riemann sum — height at the left endpoint of each subinterval
• Right Riemann sum — height at the right endpoint
• Midpoint Riemann sum — height at the middle of each subinterval (usually the most accurate of the three)
• Trapezoidal rule — averages left and right, forming trapezoids instead of rectangles

The definite integral is the limit of these sums as n → ∞:

∫ from a to b of f(x) dx = lim(n→∞) Σ f(xᵢ) Δx

A practical note: if f is increasing on [a, b], a left sum underestimates the integral and a right sum overestimates it. Reversing that logic for decreasing functions is a favorite multiple-choice question.

The Fundamental Theorem of Calculus

If Unit 6 has a single center of gravity, it’s the Fundamental Theorem of Calculus (FTC). Both parts show up constantly.

FTC Part 1 (derivatives of accumulation functions): If F(x) = ∫ from a to x of f(t) dt, then F′(x) = f(x). With the chain rule, d/dx [∫ from a to g(x) of f(t) dt] = f(g(x)) · g′(x).

FTC Part 2 (evaluating definite integrals): If F is any antiderivative of f, then ∫ from a to b of f(x) dx = F(b) − F(a).

Part 2 is what lets you evaluate integrals exactly rather than approximating them. Part 1 is what lets you differentiate functions defined as integrals — a skill tested through problems like finding where an accumulation function has a maximum or minimum. Khan Academy has a solid sequence on both parts if you want extra reps.

Worked example

Evaluate ∫ from 1 to 4 of (2x + 1) dx.

Antiderivative: F(x) = x² + x.

F(4) = 16 + 4 = 20; F(1) = 1 + 1 = 2.

F(4) − F(1) = 20 − 2 = 18.

You can sanity-check this with a trapezoidal approximation on [1, 4] with a few subintervals and see it land near 18.

Accumulation Functions and Net Change

An accumulation function has the form F(x) = ∫ from a to x of f(t) dt. The key idea: F(x) measures the net amount accumulated from a to x, given a rate f.

The net change theorem is the practical payoff: ∫ from a to b of f′(x) dx = f(b) − f(a). In words, integrating a rate of change over an interval gives the total change in the quantity.

This is why Unit 6 problems are so often dressed up as word problems:

• If r(t) is a rate of water flow in liters per minute, ∫ from 0 to 10 of r(t) dt gives total liters.
• If v(t) is velocity, ∫ v(t) dt gives displacement, while ∫ |v(t)| dt gives total distance traveled.
• If P′(t) is a population growth rate, ∫ P′(t) dt gives the population change.

Watch the distinction between displacement and distance traveled — it’s one of the most commonly missed points on the unit test. Displacement allows negative contributions to cancel; distance traveled does not.

Average value

The average value of f on [a, b] is (1/(b − a)) ∫ from a to b of f(x) dx. Geometrically, it’s the height of the rectangle over [a, b] with the same area as the region under the curve. The Mean Value Theorem for integrals guarantees that a continuous f actually attains this average value somewhere on the interval.

Indefinite Integrals and Antiderivative Rules

Indefinite integrals drop the bounds and add “+ C.” The constant matters — omitting it is a classic point deduction.

Core antiderivatives to have memorized cold:

Two properties make definite integrals easier to manipulate: ∫ from a to a of f = 0, and ∫ from a to b of f = −∫ from b to a of f. You’ll also use linearity — integrals split across sums and constant multiples — constantly.

U-Substitution for Composite Integrals

U-substitution reverses the chain rule. When an integrand contains a composite function whose inner derivative is also present (up to a constant), substitute u for the inner function.

Steps:

1. Choose u, usually the inside of a composition.
2. Compute du = u′ dx.
3. Rewrite the entire integral in terms of u and du.
4. Integrate.
5. Substitute back, or change the bounds if working with a definite integral.

Worked example

Evaluate ∫ 2x · cos(x²) dx.

Let u = x², so du = 2x dx. The integral becomes ∫ cos(u) du = sin(u) + C = sin(x²) + C.

For definite integrals, you can either convert the bounds to u-values or substitute back before evaluating. Both are accepted; pick whichever reduces errors for you.

Common Mistakes to Avoid

Most lost points at the end of this unit trace back to a handful of recurring errors:

• Dropping “+ C” on indefinite integrals.
• Confusing displacement with total distance in motion problems.
• Forgetting the chain rule when differentiating accumulation functions (FTC Part 1).
• Mishandling absolute value in ∫|v(t)| dt — you often have to split the integral at the zeros of v.
• Sign errors in FTC Part 2, especially with negative lower bounds.
• Skipping the substitution back to x when the question asks for an answer in terms of x.

A quick self-check: after computing a definite integral, ask whether the sign and magnitude make sense given the function’s graph. If f is positive throughout [a, b], the integral must be positive.

How to Prepare Efficiently

The end-of-unit checkpoint is less about new content and more about fluency. Three moves help most:

• Redo the FTC problems until they’re automatic. Both parts, with chain rule variations.
• Practice Riemann sum setups by hand. Write Δx, list the sample points, and build the sum before reaching for a calculator. The College Board includes Riemann sum questions on both calculator and non-calculator sections, so the setup skill matters even when computation is allowed.
• Mix in u-substitution with definite integrals. Choosing between changing bounds and back-substituting should be a fast decision.

Target short, frequent practice over long cramming sessions. Twenty minutes of mixed problems a day beats a single three-hour review.

Key Takeaways

• Unit 6 closes the loop between derivatives and integrals through the Fundamental Theorem of Calculus, and it accounts for a substantial share of the AP Calculus AB exam.
• Know all four Riemann sum types and which direction they over- or underestimate for monotonic functions.
• FTC Part 1 differentiates accumulation functions; FTC Part 2 evaluates definite integrals as F(b) − F(a).
• The net change theorem turns rate functions into total change — but distinguish displacement from distance traveled.
• Memorize core antiderivatives, always include “+ C” on indefinite integrals, and use u-substitution for composite integrands.
• Common errors — dropped constants, sign mistakes, and forgotten chain rules — are avoidable with a consistent post-computation sanity check.