Integration & Accumulation of Change

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 6. This unit is worth 15–20% 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 and the Definite Integral15 min · 3 objectivesCompute left, right, midpoint and trapezoidal approximations from a formula or a table · Decide whether an approximation over- or underestimates from monotonicity and concavity · Express a definite integral as the limit of a Riemann sum
• The Fundamental Theorem of Calculus15 min · 3 objectivesState both parts of the Fundamental Theorem of Calculus · Differentiate accumulation functions, including with the chain rule · Evaluate definite integrals using antiderivatives
• Accumulation Functions and the Second FTC15 min · 3 objectivesDifferentiate an accumulation function, including one with a variable expression as its upper limit · Analyze g(x) = ∫ₐˣ f(t) dt for increase, decrease, extrema and concavity using the graph of f · Evaluate an accumulation function at a point by reading areas off the graph of f
• U-Substitution and Changing the Bounds14 min · 3 objectivesChoose a substitution by looking for an inner function whose derivative is present as a factor · Convert the limits of integration when substituting in a definite integral · Recognize when substitution alone will not work and another technique is needed
• Integration by Parts15 min · 3 objectivesApply the integration-by-parts formula to products of functions · Choose u and dv strategically using the LIATE guideline · Handle repeated integration by parts and cyclic cases
• Partial Fractions14 min · 3 objectivesDecompose a rational function into a sum of simpler fractions · Solve for the unknown constants in a decomposition · Integrate the resulting terms, typically to logarithms
• Improper Integrals14 min · 3 objectivesDefine improper integrals over infinite intervals and at discontinuities · Evaluate improper integrals using limits · Determine whether an improper integral converges or diverges

Formulas in Unit 6

Every term in Unit 6

All 42 terms we publish for Integration & Accumulation of Change, 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.

Integration by parts

∫u dv = uv − ∫v du. Choose u by LIATE — logarithmic, inverse trig, algebraic, trigonometric, exponential — taking the earliest type as u.

Partial fraction decomposition

For proper rational functions with factorable denominators: split into A/(x − a) + B/(x − b), solve for constants, integrate to logarithms.

Repeated integration by parts

Needed when one application still leaves a product. Tabular integration organizes the work when u differentiates to zero.

Integration by parts returning the original

For ∫eˣ sin x dx the original integral reappears; solve algebraically for it rather than continuing.

When partial fractions applies

The numerator degree must be lower than the denominator degree. Otherwise divide first, then decompose the remainder.

Improper integral with an infinite limit

Replace the infinity with b, integrate, then take the limit as b→∞. Converges if that limit is finite.

Improper integral with a discontinuity

Split at the discontinuity and take one-sided limits. Integrating straight through an infinite discontinuity gives a wrong answer that often looks reasonable.

Trigonometric substitution qualitatively

For √(a² − x²) use x = a sin θ; for √(a² + x²) use x = a tan θ. Beyond the AB toolkit, and it converts a radical into a trig identity.

Fundamental Theorem of Calculus part 1

d/dx ∫ₐˣ f(t)dt = f(x); with a variable upper limit g(x), multiply by g′(x).

Fundamental Theorem of Calculus part 2

∫ₐᵇ f = F(b) − F(a) for any antiderivative F.

u-substitution with definite limits

Either convert the limits to u values or convert back to x before evaluating. Mixing the two is a common error.

Riemann sum approximations

Left sums underestimate an increasing function and right sums overestimate it; the trapezoid rule overestimates a concave-up one.

Net change theorem

∫ₐᵇ f′(x)dx = f(b) − f(a): integrating a rate gives the total change.

Interpreting an integral in context

State what accumulated, over what interval, with units — "8 liters of water entered the tank between t = 0 and t = 4 minutes".

Technique decision order

Substitution first, then parts, then partial fractions. Trying them out of order wastes the most time on parts, which takes longest to abandon.

LIATE

Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential — choose u as the earliest type present.

Why LIATE works

Logs and inverse trig simplify when differentiated and are painful to integrate, so they belong in u; exponentials integrate as easily as they differentiate.

Check substitution before committing to parts

∫x·e^(x²)dx looks like parts and is a one-line substitution, because 2x is present up to a constant.

Partial fractions preconditions

Denominator factors into distinct linear factors AND numerator degree is lower. Otherwise divide first.

Solving for A and B fastest

Clear denominators, then substitute the roots one at a time — each substitution kills one unknown.

Partial fractions produce logarithms

∫A/(x − a)dx = A·ln|x − a| + C. The absolute values matter where the argument is negative.

Outside the AP scope

Repeated factors and irreducible quadratics are not tested. Encountering one means the intended method was something else.

Improper: infinite interval

∫₁^∞ f means lim(b→∞) ∫₁^b f. Write the limit — the rubric awards it separately.

Improper: infinite integrand

A discontinuity inside the interval also makes an integral improper; take the limit approaching the bad point.

Never substitute ∞

Writing "−1/(2·∞²)" loses the setup point even when the final number is right. ∞ is not a value.

p-integral at infinity

∫₁^∞ dx/xᵖ converges exactly when p > 1, with value 1/(p − 1).

p-integral near zero

∫₀¹ dx/xᵖ converges exactly when p < 1 — the condition reverses, because the trouble is at the origin.

At infinity shrink fast, near zero blow up slowly

The mnemonic that keeps the two p-integral conditions straight.

What the integral test does not give

It settles convergence only. When both converge the sum and the integral are NOT equal.

FTC Part 2 with a chain rule

d/dx ∫ₐ^(u(x)) f = f(u(x))·u′(x). Substitute the upper limit, then multiply by its derivative.

Variable in the lower limit

The sign flips: d/dx ∫ₓ^b f = −f(x), because swapping limits negates an integral.

Do not integrate when asked to differentiate

Part 2 exists so you need not. The exam includes integrands with no elementary antiderivative to enforce this.

g′ = f and g″ = f′

For g(x) = ∫ₐˣ f, every Unit 5 question about f becomes a question about g shifted one derivative.

g(a) = 0 always

The integral from a to a is zero, and this is often the only value of g supplied.

Left sum with an increasing function

Underestimates. Monotonicity — not concavity — decides left and right sums.

Trapezoid and concavity

Concave up puts each chord above the curve, so the trapezoidal sum overestimates. Concave down reverses it.

Midpoint runs opposite the trapezoid

On a concave-up function the midpoint rule underestimates while the trapezoid overestimates.

Unequal widths are normal

Table data rarely has equal spacing, and the (h/2) shortcut is valid only when it does.

Reversal property

∫ᵇᵃ f = −∫ₐᵇ f. The property most often forgotten, and the one recombination questions are built on.

Integral of a constant

∫ₐᵇ c dx = c(b − a). The most-dropped term when recombining given integral values.

No product rule for integrals

∫fg is not ∫f·∫g. If a question appears to need that, the intended method is substitution.

u-substitution changes the limits

Convert the limits to u values, or convert back to x before evaluating. Mixing them is a silent error.

What examiners penalize here

• Riemann-sum questions on the calculator-free section reward writing the sum before evaluating it: the setup line carries its own point, and an arithmetic slip afterward costs less than a missing setup.
• FTC Part 1 questions on the AP exam almost always disguise a chain-rule step by using a non-trivial upper limit. Before answering, check whether the upper limit is plain x or a function of x — the latter needs the extra u′(x) factor.
• The graph-of-f accumulation question appears on nearly every exam and carries several points at once: values of g, an extremum with justification, concavity, and often a tangent line to g. Getting g′ = f stated early sets up all of them.
• BC Unit 6 is the largest unit on the exam, and much of its weight is technique selection rather than execution. When practicing, say which tool you are choosing and why before doing any algebra.
• Some integrals like ∫eˣ·sin(x) dx cycle back to the original after two rounds of parts. When that happens, set the integral equal to I, solve the algebraic equation for I, and divide — do not keep integrating forever.
• The fastest way to find the constants is the "cover-up" method: to get the constant over (x − a), cover that factor and substitute x = a into what remains. It computes each numerator in one step, ideal under time pressure.
• Improper integrals also arise from a vertical asymptote *inside* or at an endpoint of the interval, such as ∫₀¹ 1/√x dx. Split at the discontinuity and take a one-sided limit there — do not treat it as an ordinary definite integral.

Practice Calculus BC

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 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series