AP Calculus AB Unit 6: Integration as Accumulation (Riemann Sums, FTC, Numerical Methods, Substitution, and Symmetry)

Accumulation

The process of adding up many small contributions (often given by a rate) to find a total change over an interval; a core meaning of the definite integral.

Rate of change

How fast a quantity changes with respect to another variable; written as a derivative like Q'(t)=dQ/dt.

Definite integral

An accumulation from a to b written ∫_a^b f(x) dx; interpreted as signed area or net change, depending on context.

Signed area

Area above the x-axis counted positive and area below counted negative; ∫_a^b f(x) dx equals signed area between f and the x-axis on [a,b].

Net change

Overall change that includes increases and decreases with sign; often represented by a definite integral of a rate.

Total change (total amount)

Amount accumulated ignoring sign (e.g., total distance or total area); often requires absolute value rather than a standard definite integral.

Net Change Theorem

If F'(x)=f(x), then F(b)-F(a)=∫_a^b f(x) dx; integrals of rates give accumulated change.

Displacement

Net change in position; computed as ∫_a^b v(t) dt where v(t) is velocity (can be zero even if motion occurred).

Flow rate

A rate describing volume per unit time (e.g., gallons/min); integrating it over time gives total volume added.

Riemann sum

A finite sum approximating an integral: Σ{i=1}^n f(xi*)Δx, representing accumulated rectangles’ areas.

Partition

Breaking an interval [a,b] into subintervals to build a Riemann sum or numerical approximation.

Subinterval

One piece of a partition, typically [x{i-1}, xi], over which a rectangle/trapezoid approximation is formed.

Δx (equal subinterval width)

For n equal subintervals on [a,b], Δx=(b−a)/n; the width of each rectangle/trapezoid.

Sample point (x_i*)

A chosen x-value in each subinterval used to set the rectangle height f(x_i*) in a Riemann sum.

Left Riemann sum (L_n)

A Riemann sum using left endpoints as sample points: xi*=x{i-1}.

Right Riemann sum (R_n)

A Riemann sum using right endpoints as sample points: xi*=xi.

Midpoint Riemann sum

A Riemann sum using midpoints as sample points: xi*=(x{i-1}+x_i)/2.

Definite integral as a limit of sums

Definition: ∫a^b f(x) dx = lim{n→∞} Σ{i=1}^n f(xi*)Δx.

Dummy variable

A variable inside an integral that has no meaning outside it (e.g., ∫_a^x f(t) dt); it can be renamed without changing value.

Limits of integration

The bounds a and b in ∫_a^b f(x) dx that specify the interval of accumulation.

Integrand

The function being accumulated in an integral, e.g., f(x) in ∫_a^b f(x) dx.

Differential (dx)

Indicates the variable of integration and the tiny “width” of slices; important for substitution and unit analysis.

Reversing limits property

Switching bounds changes the sign: ∫a^b f(x) dx = −∫b^a f(x) dx.

Zero-length interval property

An integral over an interval of length 0 is 0: ∫_a^a f(x) dx = 0.

Additivity over intervals

You can split an interval: ∫a^b f(x) dx = ∫a^c f(x) dx + ∫_c^b f(x) dx.

Linearity of integrals

Integrals distribute over addition and constants: ∫(f+g)=∫f+∫g and ∫(kf)=k∫f (with same bounds).

Comparison property

If f(x)≥g(x) on [a,b], then ∫a^b f(x) dx ≥ ∫a^b g(x) dx.

Accumulation function

A function defined by integrating to a variable endpoint: A(x)=∫_a^x f(t) dt; it measures accumulated amount from a to x.

Fundamental Theorem of Calculus (Part 1)

If f is continuous, then d/dx(∫_a^x f(t) dt)=f(x); differentiating an accumulation function returns the integrand.

Chain rule with variable upper limit (FTC Part 1 general form)

d/dx(∫_a^{g(x)} f(t) dt)=f(g(x))·g'(x).

Increasing/decreasing of an accumulation function

For A(x)=∫_a^x f(t) dt, A'(x)=f(x), so A increases where f>0 and decreases where f<0.

Local extrema of an accumulation function

A(x)=∫_a^x f(t) dt has extrema where f(x)=0 with a sign change (positive→negative gives local max; negative→positive gives local min).

Concavity of an accumulation function

For A(x)=∫_a^x f(t) dt, A''(x)=f'(x); A is concave up where f is increasing and concave down where f is decreasing.

Antiderivative

A function F such that F'(x)=f(x); used to evaluate definite integrals via FTC Part 2.

Indefinite integral

Notation for the family of antiderivatives: ∫ f(x) dx = F(x)+C (not a single number).

Constant of integration (+C)

A constant added to an indefinite integral because derivatives lose constant information; all antiderivatives differ by a constant.

Power rule for antiderivatives

For n≠−1: ∫ x^n dx = x^{n+1}/(n+1) + C; often requires algebraic rewriting first.

Fundamental Theorem of Calculus (Part 2 / Evaluation Theorem)

If F'(x)=f(x), then ∫_a^b f(x) dx = F(b)−F(a); computes definite integrals from antiderivatives.

Numerical integration

Approximating a definite integral using sums/shapes (rectangles or trapezoids) when a formula or antiderivative is unavailable.

Midpoint Rule (M_n)

A numerical method using midpoints: Mn=Δx Σ{i=1}^n f((x{i-1}+xi)/2) for equal subintervals.

Trapezoidal Rule (T_n)

A numerical method using trapezoids: Tn=(Δx/2)[f(x0)+2f(x1)+…+2f(x{n-1})+f(x_n)] for equal subintervals.

Left/right sum over/under behavior (monotonicity)

If f is increasing on [a,b], left sums tend to underestimate and right sums tend to overestimate the integral (reversed if f is decreasing).

Trapezoidal rule error direction (concavity)

If f is concave up, Tn tends to overestimate; if f is concave down, Tn tends to underestimate.

u-substitution

Integration technique that undoes the chain rule by substituting u=g(x) to simplify an integrand with a composition.

Reverse chain rule pattern

If an integrand matches f'(g(x))g'(x), then an antiderivative is f(g(x))+C.

Changing bounds in u-substitution

For definite integrals, you may change x-limits to u-limits after substituting; do not mix changed bounds with back-substitution inconsistently.

Even function

A function with symmetry about the y-axis: f(−x)=f(x).

Odd function

A function with origin symmetry: f(−x)=−f(x).

Symmetry shortcuts on [−a,a]

If f is even, ∫{−a}^a f(x) dx = 2∫0^a f(x) dx; if f is odd, ∫_{−a}^a f(x) dx = 0.

Basic trig antiderivative pair (cos/sin)

Since d/dx(sin x)=cos x, it follows that ∫ cos(x) dx = sin(x) + C (a common memorized rule).