Integration and Accumulation of Change

Fourteen topics that build the inverse of the derivative. Accumulation from a rate graph, with net change against total area, Riemann sums and whether they over- or under-estimate, sigma and integral notation, the Fundamental Theorem in both directions, the properties that let you rearrange without evaluating, antiderivatives and the constant an initial condition pins down, substitution and the limits it changes, long division and completing the square, and, for BC, parts, partial fractions and improper integrals.

AB exam 15-20%BC exam 15-20%14 topics

Topics

6.1

Exploring Accumulations of Change

Approximating Areas with Riemann Sums

Riemann Sums, Summation Notation, and Definite Integral Notation

The Fundamental Theorem of Calculus and Accumulation Functions

Interpreting the Behavior of Accumulation Functions Involving Area

Applying Properties of Definite Integrals

The Fundamental Theorem of Calculus and Definite Integrals

Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Integrating Using Substitution

Integrating Functions Using Long Division and Completing the Square

Integrating Using Integration by PartsBC only

Integrating Using Linear Partial FractionsBC only

Evaluating Improper IntegralsBC only

Selecting Techniques for Antidifferentiation

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Area under a rate

an AMOUNT; units are the two axes multiplied

Net change

SIGNED area: below the axis subtracts

Total area

both pieces counted positively; a different question

Subinterval width

Δx = (b − a)/n, and there are n TERMS

Left vs right sums

decided by whether f is INCREASING

Trapezoid

decided by CONCAVITY; midpoint leans the other way

Definite integral

the limit of those sums; a NUMBER

FTC Part 1

d/dx of ∫ f from a to u(x) = f(u(x)) · u′(x)

Variable LOWER limit

flips the sign; both moving gives two terms

FTC Part 2

∫ f from a to b = F(b) − F(a), top minus bottom

Net Change Theorem

∫ of a RATE is a CHANGE; add the starting amount

Reading g from f

g′ = f and g″ = f′: sign and turns, never size

Reversal

swapping the limits NEGATES

Additivity

split at c; pieces must meet without overlap or gap

Linearity

sums and CONSTANT multiples split

Products and quotients

NEVER split. There is no such property

Symmetry

on [−a, a]: odd gives 0, even doubles the half

Power rule, reversed

raise, then divide by the NEW exponent; n ≠ −1

The excluded case

∫ dx/x = ln|x| + C, bars included

Linear inside

a reciprocal coefficient comes out front

Constant of integration

indefinite YES, definite NO; a point determines it

Substitution

every x must go; convert the limits OR convert back

Improper fraction

DIVIDE first; irreducible quadratic COMPLETES

Parts (BC)

∫ u dv = uv − ∫ v du; the new one must be easier

Partial fractions (BC)

one CONSTANT over each distinct linear factor

Improper integrals (BC)

write the LIMIT; split at a blow-up inside

Choosing (6.14)

basic form, then rewrite, then substitute, then parts

Unit 6 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 14 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from. AB students are served only the AB items; BC students get the whole unit.

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Check what stuck.

Units 1 through 6, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

20 or 45questions

62topics

88codes covered

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