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
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