Unit 6 (AP Calculus BC): Integration and Accumulation of Change — Comprehensive Study Notes

Rate of change

A quantity describing how fast something changes per unit of an input (e.g., meters per second).

Accumulated change

The total change over an interval found by adding up many small changes; modeled by a definite integral.

Integral

A calculus operation that represents accumulation; for definite integrals, it measures net accumulation over an interval.

Antiderivative

A function F whose derivative is f; i.e., F'(x)=f(x).

Definite integral

An integral with bounds a and b, ∫_a^b f(x) dx, representing net accumulation (signed area) over [a,b].

Indefinite integral

The family of all antiderivatives of f: ∫ f(x) dx = F(x) + C.

Constant of integration (C)

A constant added to an indefinite integral because derivatives of constants are zero.

Signed area

Area above the x-axis counted positive and area below the x-axis counted negative; equals the value of a definite integral.

Net change

Total accumulated change that includes sign/direction (negative rates subtract from the total).

Total change (total amount)

Total magnitude accumulated regardless of direction; typically computed using absolute value and splitting at sign changes.

Displacement

Net change in position; equals ∫_a^b v(t) dt when v(t) is velocity.

Total distance traveled

Total amount of motion; equals ∫_a^b |v(t)| dt (requires splitting where v(t)=0).

Speed

The magnitude of velocity; speed = |v(t)|.

Units check

Using dimensional analysis to verify an integral setup (rate units × time units = accumulated units).

Riemann sum

An approximation of a definite integral by adding rectangle areas: Σ f(x_i*)Δx.

Partition

A division of an interval [a,b] into subintervals used to build a Riemann sum.

Subinterval width (Δx)

For equal subintervals, Δx = (b−a)/n; the base length of each rectangle/trapezoid.

Sample point (x_i*)

A chosen x-value in each subinterval used to determine the rectangle height f(x_i*).

Left Riemann sum

A Riemann sum using left endpoints of subintervals as sample points.

Right Riemann sum

A Riemann sum using right endpoints of subintervals as sample points.

Midpoint Riemann sum

A Riemann sum using midpoints of subintervals as sample points.

Overestimate

An approximation larger than the true integral value (e.g., right sum for an increasing function).

Underestimate

An approximation smaller than the true integral value (e.g., left sum for an increasing function).

Definite integral as a limit

The formal definition: ∫a^b f(x) dx = lim{n→∞} Σ f(x_i*)Δx.

Sigma notation (Σ)

A compact notation for sums, commonly used to write Riemann sums and numerical integration formulas.

Right-endpoint formula for x_i

For equal partitions, right endpoints are x_i = a + iΔx (i=1 to n).

Left-endpoint formula for x_i

For equal partitions, left endpoints are x_i = a + (i−1)Δx (i=1 to n).

Trapezoidal Rule

Numerical integration using trapezoids: Tn = (Δx/2)[f(x0)+2f(x1)+…+2f(x{n−1})+f(x_n)].

Interior nodes (Trapezoidal Rule)

The points x1 through x{n−1} that receive coefficient 2 in T_n because each is shared by two trapezoids.

Concave up

A graph curving upward (f''>0); trapezoidal approximations tend to overestimate for concave-up functions.

Concave down

A graph curving downward (f''<0); trapezoidal approximations tend to underestimate for concave-down functions.

Simpson’s Rule

Numerical integration using parabolas over pairs of subintervals: Sn = (Δx/3)[f(x0)+4f(x1)+2f(x2)+…+4f(x{n−1})+f(xn)].

Even number of subintervals (Simpson’s Rule requirement)

Simpson’s Rule requires n to be even so the interval can be grouped into pairs of subintervals.

Fundamental Theorem of Calculus (FTC) Part 1

If F(x)=∫_a^x f(t) dt and f is continuous, then F'(x)=f(x).

Accumulation function

A function defined by a variable-limit integral, such as F(x)=∫_a^x f(t) dt, representing accumulated total from a to x.

FTC Part 1 chain rule (variable upper limit)

If G(x)=∫_a^{g(x)} f(t) dt, then G'(x)=f(g(x))·g'(x).

Variable lower limit rule

If H(x)=∫_{h(x)}^a f(t) dt, then H'(x)=−f(h(x))·h'(x).

Increasing/decreasing test for accumulation functions

If F(x)=∫_a^x f(t) dt, then F increases where f(x)>0 and decreases where f(x)<0.

Concavity test for accumulation functions

If F(x)=∫_a^x f(t) dt, then F''(x)=f'(x), so F is concave up where f is increasing and concave down where f is decreasing.

Fundamental Theorem of Calculus (FTC) Part 2 (Evaluation Theorem)

If F'(x)=f(x) on [a,b], then ∫_a^b f(x) dx = F(b) − F(a).

Bracket notation [F(x)]_a^b

A notation meaning F(b) − F(a), used to evaluate definite integrals via antiderivatives.

Power rule for integrals (n ≠ −1)

∫ x^n dx = x^{n+1}/(n+1) + C.

Linearity of integrals

∫a^b (cf+dg) dx = c∫a^b f dx + d∫_a^b g dx.

Additivity over intervals

If a<b<c, then ∫a^c f dx = ∫a^b f dx + ∫_b^c f dx.

Reversing bounds property

∫b^a f(x) dx = −∫a^b f(x) dx.

Symmetry (even function integral)

If f is even, then ∫{−a}^a f(x) dx = 2∫0^a f(x) dx.

Symmetry (odd function integral)

If f is odd, then ∫_{−a}^a f(x) dx = 0.

Average value of a function

The average of f on [a,b]: favg = (1/(b−a))∫a^b f(x) dx.

Mean Value Theorem for Integrals

If f is continuous on [a,b], there exists c in [a,b] such that f(c) = (1/(b−a))∫_a^b f(x) dx.

U-substitution

An integration technique that reverses the chain rule by substituting u=g(x) to simplify the integrand before integrating.