AP Calculus AB/BC Unit 6 ReviewIntegration and Accumulation of Change

AP Calculus Unit 6, Integration and Accumulation of Change, is where the second half of calculus begins. If a derivative tells you how fast something is changing at one instant, an integral adds up all that change over an interval, and the single biggest idea here is the Fundamental Theorem of Calculus, which proves that differentiation and integration are inverse operations. The unit runs from approximating area with Riemann sums to evaluating definite integrals exactly with antiderivatives, plus a toolbox of techniques like u-substitution (and integration by parts, partial fractions, and improper integrals on BC). It is worth 17-20% of the AP exam, the largest weight of any unit on the AB exam.

What this unit covers

From area to the definite integral

• The area between a rate-of-change graph and the x-axis is the accumulation of change. If $f(t)$ is in gallons per minute and $t$ is in minutes, the area under $f$ is measured in gallons. Units multiply, which is how you interpret these areas in context.
• Area above the x-axis counts as positive accumulated change; area below counts as negative. That sign convention is why a definite integral can be zero even when plenty is happening.
• Riemann sums approximate that area with rectangles. You need left, right, and midpoint sums plus trapezoidal sums, and you have to handle nonuniform partitions, which is exactly what table-based exam problems give you.
• Whether a sum overestimates or underestimates depends on the function's behavior. A left sum underestimates an increasing function; a trapezoidal sum overestimates a concave-up one. Expect to justify these.
• The definite integral is defined as the limit of Riemann sums as the subinterval widths shrink to zero. You should be able to translate between summation notation, $\lim_{\max \Delta x_i \to 0} \sum f(x_i^*)\,\Delta x_i$, and integral notation $\int_a^b f(x)\,dx$, in both directions.

The Fundamental Theorem of Calculus

• One part of the FTC says that if $g(x) = \int_a^x f(t)\,dt$, then $g'(x) = f(x)$. The definite integral builds new functions (accumulation functions), and differentiating one hands you back the integrand.
• The other part is the evaluation tool. If $F$ is any antiderivative of a continuous $f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. This is what lets you skip Riemann sums entirely when a closed-form antiderivative exists.
• Accumulation functions inherit behavior from their integrand. Where $f$ is positive, $g(x) = \int_a^x f(t)\,dt$ is increasing; where $f$ crosses from positive to negative, $g$ has a maximum; where $f$ is increasing, $g$ is concave up. Reading a graph of $f$ and describing $g$ is one of the most common FRQ setups in the entire course.
• Properties of definite integrals let you compute with pieces. You can pull out constants, split sums, reverse limits (which flips the sign), and stitch adjacent intervals together with $\int_a^b + \int_b^c = \int_a^c$. Many problems give you a graph made of lines and semicircles and expect you to evaluate integrals with pure geometry.
• The definition extends to functions with removable or jump discontinuities, so a single hole or step does not break integrability.

Antiderivatives and basic integration

• An antiderivative of $f$ is any function $F$ with $F' = f$. The indefinite integral $\int f(x)\,dx = F(x) + C$ collects all of them, and the $+C$ is required, not decorative.
• Every derivative rule you learned runs in reverse here. The power rule becomes $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$, and you need the antiderivatives of $\sin x$, $\cos x$, $\sec^2 x$, $e^x$, $1/x$, and the inverse-trig forms like $\frac{1}{1+x^2}$.
• Some functions, like $e^{-t^2}$, have no closed-form antiderivative at all. That is precisely why accumulation functions matter; $\int_0^x e^{-t^2}\,dt$ is a perfectly good function even though you cannot write its antiderivative with elementary functions.

Techniques for harder integrands

• U-substitution is the chain rule in reverse. Pick $u = g(x)$ so that $du = g'(x)\,dx$ appears in the integrand, rewrite, and integrate. For definite integrals, change the limits to u-values too, or convert back to x before plugging in. Mixing those up is a classic point-loser.
• Long division handles rational functions where the numerator's degree is at least the denominator's. Divide first, then integrate the polynomial part and the remainder separately.
• Completing the square turns quadratics like $x^2 + 4x + 13$ into $(x+2)^2 + 9$, which sets up an arctangent antiderivative.
• Integration by parts (BC only) reverses the product rule with $\int u\,dv = uv - \int v\,du$. Use it for products like $x e^x$ or $x \ln x$.
• Linear partial fractions (BC only) decompose rational functions with distinct linear factors, like $\frac{1}{(x-1)(x+2)}$, into simple fractions that integrate to logarithms. This technique pays off again with the logistic equation in Unit 7.
• Improper integrals (BC only) have an infinite limit of integration or an unbounded integrand. You evaluate them as limits of definite integrals and decide whether they converge to a value or diverge.
• The final skill is choosing the right tool. Before computing anything, ask whether the integrand needs basic rules, substitution, algebraic rearrangement, parts, or partial fractions. The exam tests this judgment directly.

Unit 6, Integration and Accumulation of Change at a glance

Why Unit 6, Integration and Accumulation of Change matters in AP Calc

This unit completes the course's central pair of ideas. Units 1 through 5 built the derivative as instantaneous rate of change; Unit 6 builds the integral as accumulated change, and the FTC welds them together into one structure. Everything after this point in the course is an application of what you learn here.

• It carries the heaviest exam weight in the course at 17-20%, so fluency here pays off more per topic than anywhere else.
• The FTC is the payoff of the limits idea from the start of the course. A definite integral is literally a limit of Riemann sums, so the analysis-of-functions theme and the limits theme converge in this unit.
• Antidifferentiation is the engine for Units 7 and 8. You cannot solve a differential equation or compute a volume without the techniques from topics 6.8 through 6.10.
• Accumulation-function analysis recycles all of Unit 5's logic (increasing/decreasing, extrema, concavity) in integral form, which is why graph-of-f FRQs are so common.

How this unit connects across the course

• The definite integral is defined as a limit (Unit 1), and the FTC's proof relies on continuity. The "limit of Riemann sums" definition is the same limiting idea you used to define the derivative.
• Every antiderivative rule reverses a derivative rule (Units 2 and 3). U-substitution is the chain rule backwards, integration by parts is the product rule backwards, and the inverse-trig antiderivatives come straight from Unit 3's inverse-function derivatives.
• Analyzing an accumulation function $g(x) = \int_a^x f(t)\,dt$ reuses the full Unit 5 toolkit. The graph of $f$ plays the role the graph of $f'$ played there, so the reasoning about extrema and concavity transfers directly.
• Solving separable differential equations (Unit 7) is antidifferentiation with initial conditions, and on BC, partial fractions from this unit unlock the logistic model.
• Areas between curves, volumes, and average value (Unit 8) all assume you can set up and evaluate definite integrals without hesitation. On BC, improper integrals return in Unit 10 as the integral test for series convergence.

Key formulas and procedures

• $\int_a^b f(x)\,dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^{n} f(x_i^*)\,\Delta x_i$ defines the definite integral as a limit of Riemann sums.
• Riemann sum setup: each term is (function value) × (subinterval width); for nonuniform partitions from a table, compute each width separately.
• Trapezoidal sum: average the left and right heights on each subinterval, $\frac{f(x_{i-1}) + f(x_i)}{2}\,\Delta x_i$, then add.
• FTC, derivative form: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$; with a function upper limit, chain rule gives $\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x))\cdot u'(x)$.
• FTC, evaluation form: $\int_a^b f(x)\,dx = F(b) - F(a)$ whenever $F' = f$.
• Power rule for antiderivatives: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$, and $\int \frac{1}{x}\,dx = \ln|x| + C$.
• Integral properties: $\int_a^b c f = c\int_a^b f$, $\int_a^b (f+g) = \int_a^b f + \int_a^b g$, $\int_b^a f = -\int_a^b f$, and $\int_a^b f + \int_b^c f = \int_a^c f$.
• U-substitution: set $u = g(x)$, $du = g'(x)\,dx$, and for definite integrals convert the limits to $u(a)$ and $u(b)$.
• Long division and completing the square rewrite rational and quadratic integrands into integrable forms (polynomials plus log or arctan pieces).
• Integration by parts (BC): $\int u\,dv = uv - \int v\,du$; choose $u$ as the factor that simplifies when differentiated.
• Partial fractions (BC): write $\frac{p(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$, solve for A and B, integrate to logarithms.
• Improper integrals (BC): replace the problem bound with a variable and take a limit, e.g. $\int_1^{\infty} f(x)\,dx = \lim_{b \to \infty}\int_1^b f(x)\,dx$; state convergence or divergence.

Unit 6, Integration and Accumulation of Change on the AP exam

At 17-20% of the exam, this is the single most heavily weighted unit on the AB exam and a major share of BC. Multiple-choice questions test antiderivative rules, u-substitution (often with limit changes built into the wrong answers), FTC derivatives of accumulation functions, and integral properties where you combine given values like $\int_0^5 f = 7$ to find related integrals. BC multiple choice adds parts, partial fractions, and convergence of improper integrals.

In the free-response section, this unit anchors two recurring problem types. The first is the table problem, where you approximate a definite integral with a left, right, midpoint, or trapezoidal sum from data, interpret the result with units in context, and often state whether your approximation is an over- or underestimate with a reason. The second is the graph-of-f problem, where $g(x) = \int_a^x f(t)\,dt$ is defined from a piecewise graph and you find values of $g$ using geometry, locate extrema and inflection points of $g$, and justify your answers using the sign and behavior of $f$. Both require clean communication, so practice writing justifications like "g has a relative maximum at x = 2 because g' = f changes from positive to negative there."

Essential questions

• How can adding up infinitely many infinitely thin pieces produce an exact answer rather than an approximation?
• Why does the area under a rate-of-change graph tell you the total change in the original quantity?
• In what sense are differentiation and integration inverse operations, and what does the Fundamental Theorem of Calculus actually guarantee?
• How do you recognize which antidifferentiation technique an integrand calls for?

Key terms to know

• Accumulation of change: The net total change in a quantity over an interval, found as the signed area under its rate-of-change graph.
• Riemann sum: An approximation of a definite integral built from products of function values and subinterval widths over a partition.
• Partition: The division of an interval into subintervals, which may be uniform or nonuniform, for building a Riemann sum.
• Trapezoidal sum: An approximation that uses trapezoids instead of rectangles, equivalent to averaging the left and right sums on each subinterval.
• Definite integral: The limit of Riemann sums as subinterval widths approach zero, written $\int_a^b f(x)\,dx$, which yields a number.
• Indefinite integral: The family of all antiderivatives of a function, written $\int f(x)\,dx = F(x) + C$.
• Antiderivative: A function $F$ whose derivative is the given function $f$.
• Constant of integration: The $+C$ attached to every indefinite integral because antiderivatives differ by a constant.
• Accumulation function: A function defined by a definite integral with a variable limit, like $g(x) = \int_a^x f(t)\,dt$.
• Fundamental Theorem of Calculus: The pair of results linking derivatives and integrals, giving both $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ and $\int_a^b f\,dx = F(b) - F(a)$.
• U-substitution: The reverse of the chain rule, replacing part of an integrand with a new variable to simplify it.
• Integration by parts: The reverse of the product rule, $\int u\,dv = uv - \int v\,du$ (BC only).
• Partial fraction decomposition: Rewriting a rational function as a sum of simpler fractions with linear, nonrepeating denominators (BC only).
• Improper integral: An integral with an infinite limit of integration or an unbounded integrand, evaluated as a limit and classified as convergent or divergent (BC only).

Common mix-ups

• A definite integral is not always an "area" in the everyday sense. Regions below the x-axis contribute negative values, so net change can be zero or negative even when the graph encloses visible area. Total area requires integrating $|f(x)|$.
• When you u-substitute in a definite integral, the original limits are x-values. Either convert them to u-values or convert your antiderivative back to x before evaluating. Plugging x-limits into a u-expression is one of the most common errors in the unit.
• The FTC derivative shortcut $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ only works directly when the upper limit is plain $x$. If the limit is $x^2$ or some other function, you must multiply by its derivative via the chain rule.
• An overestimate or underestimate from a Riemann sum depends on whether $f$ is increasing or decreasing (for left/right sums) or its concavity (for midpoint/trapezoidal sums), not on which sum "usually" runs high. Justify with the function's behavior on that specific interval.