Unit 6: Integration and Accumulation of Change

Accumulation

Rebuilding a total change by adding many small contributions from a rate over an interval.

Rate of change

How a quantity changes per unit of the input (e.g., velocity in m/s, flow in gal/min).

Definite integral

An integral with limits of integration that outputs a single number representing net accumulated change on an interval.

Indefinite integral

An integral without bounds that represents a family of antiderivatives, written as F(x)+C.

Antiderivative

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

Net (signed) area

Interpretation of ∫_a^b f(x) dx as area above the x-axis minus area below the x-axis on [a,b].

Net change

The total change in a quantity over an interval, often computed as the definite integral of its rate.

Total area

Area counting all regions as positive; often written as ∫_a^b |f(x)| dx.

Displacement

Net change in position: ∫_a^b v(t) dt.

Total distance traveled

Total path length from velocity: ∫_a^b |v(t)| dt.

Units check (integrals)

A sanity-check: if f has units (quantity per x-unit), then ∫ f(x) dx has units of the quantity because you multiply by dx.

Limits of integration

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

Riemann sum

An approximation to a definite integral using a sum of rectangle areas: Σ f(x_i*)Δx.

Partition

A division of [a,b] into n subintervals using points x0=a, x1, …, xn=b.

Subinterval width (Δx)

For equal partitions, Δx=(b−a)/n.

Sample point (x_i*)

A chosen input in each subinterval [x{i−1}, xi] used to define rectangle height in a Riemann sum.

Left Riemann sum (L_n)

Riemann sum using left endpoints: Σ{i=1}^n f(x{i−1})Δx.

Right Riemann sum (R_n)

Riemann sum using right endpoints: Σ{i=1}^n f(xi)Δx.

Midpoint Riemann sum (M_n)

Riemann sum using midpoints: Σ f((x{i−1}+xi)/2)Δx.

Definite integral as a limit

If the limit exists, ∫a^b f(x) dx = lim{n→∞} Σ{i=1}^n f(xi*)Δx.

Trapezoidal rule (T_n)

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

Trapezoidal rule as an average

For equal subintervals, the trapezoidal approximation satisfies Tn=(Ln+R_n)/2.

Uneven subinterval widths

When x-values are not evenly spaced, you must use each interval’s actual width instead of a single Δx.

Endpoint selection pitfall

In a left sum, do not use the rightmost function value; in a right sum, do not use the leftmost function value.

Fundamental Theorem of Calculus (FTC)

The theorem connecting differentiation and integration as inverse processes, enabling exact evaluation via antiderivatives.

FTC Part 1

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

Chain rule with variable upper limit

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

Variable lower limit sign change

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

FTC Part 2

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

Net Change Theorem

If a quantity has rate F'(x), then F(b)−F(a)=∫_a^b F'(x) dx.

Linearity of integrals

∫a^b (f+g) dx = ∫a^b f dx + ∫a^b g dx and ∫a^b c f dx = c∫_a^b f dx.

Additivity over intervals

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

Reversing bounds property

∫a^b f dx = −∫b^a f dx.

Zero-width interval property

∫_a^a f(x) dx = 0.

Even function

A function with f(−x)=f(x); on [−a,a], ∫{−a}^a f(x) dx = 2∫0^a f(x) dx.

Odd function

A function with f(−x)=−f(x); on [−a,a], ∫_{−a}^a f(x) dx = 0.

Symmetric interval requirement

Even/odd integral shortcuts apply only on intervals of the form [−a,a].

Integral comparison property

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

Bounding (min/max) property

If m ≤ f(x) ≤ M on [a,b], then m(b−a) ≤ ∫_a^b f(x) dx ≤ M(b−a).

Constant of integration (+C)

A constant added to an indefinite integral because antiderivatives differ by constants; omitted for definite integrals.

Power rule for antiderivatives (n≠−1)

∫ x^n dx = x^{n+1}/(n+1) + C, for n≠−1.

Logarithm antiderivative case

∫ (1/x) dx = ln|x| + C (the special case where n=−1).

Exponential antiderivative (e^x)

∫ e^x dx = e^x + C.

Exponential antiderivative (a^x)

∫ a^x dx = a^x/ln(a) + C, for a>0 and a≠1.

Basic trig antiderivative pair: cos

∫ cos x dx = sin x + C.

Basic trig antiderivative pair: sin

∫ sin x dx = −cos x + C.

u-substitution (integration by substitution)

A method that reverses the chain rule by setting u=g(x) to rewrite ∫ f(g(x))g'(x) dx as ∫ f(u) du.

Changing bounds in substitution

For definite integrals with u-substitution, convert x-bounds to u-bounds to evaluate entirely in u (or substitute back consistently).

Accumulation function

A function defined by an integral like A(x)=∫_a^x r(t) dt, representing accumulated net change up to x.

Initial value plus accumulated change

If Q'(t)=r(t) and Q(a) is known, then Q(b)=Q(a)+∫_a^b r(t) dt (final amount = initial + change).