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 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

• 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
• 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 14 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

Factor the denominator into distinct linear factors and split into a sum of simpler fractions, each integrating to a logarithm.

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".

What examiners penalize here

• 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.
• 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