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 $\bm{\int}_a^b f(x) \,dx$ 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 $\bm{\int}_a^b |f(x)| \,dx.$

Displacement

Net change in position: $\int_a^b v(t) dt$.

Total distance traveled

Total path length from velocity: $\bm{\int}_a^b |v(t)| \,dt.$

Units check (integrals)

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

Limits of integration

The bounds $a$ and $b$ in $\bm{\int}_a^b f(x) \,dx$ that determine the interval of accumulation.

Riemann sum

An approximation to a definite integral using a sum of rectangle areas: $\bm{\Sigma} f(x_i*)Δx.$

Partition

A division of $[a,b]$ into $n$ subintervals using points $x_0=a, x_1, \ldots, x_n=b$.

Subinterval width (Δx)

For equal partitions, Δx=(b−a)/n.

Sample point (x_i*)

A chosen input in each subinterval [$x_{i-1}, x_i$] used to define rectangle height in a Riemann sum.

Left Riemann sum (L_n)

Riemann sum using left endpoints: $\bm{\Sigma}_{i=1}^n f(x_{i-1})Δx.$

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: $\bm{\Sigma} f\bigg(\frac{(x_{i-1}+x_i)}{2}\bigg)Δx.$

Definite integral as a limit

If the limit exists, $\bm{\int}_a^b f(x) \,dx = \bm{\lim}_{n\to\infty} \bm{\Sigma}_{i=1}^n f(x_i*)Δx.$

Trapezoidal rule (T_n)

Approximation using trapezoids: $T_n=\frac{Δx}{2}\bigg[f(x_0)+2f(x_1)+…+2f(x_{n-1})+f(x_n)\bigg].$

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)=$\int_a^x f(t) dt$ and f is continuous, then F'(x)=f(x).

Chain rule with variable upper limit

If G(x)=$\int_a^{g(x)} f(t) dt$, then G'(x)=f(g(x))·g'(x).

Variable lower limit sign change

If $H(x)=\bm{\int}_{g(x)}^a f(t) \,dt$, then $H'(x)=-f(g(x))·g'(x).$

FTC Part 2

If F'(x)=f(x), then $\int_a^b f(x) dx = F(b)-F(a)$.

Net Change Theorem

If a quantity has rate $F'(x)$, then $F(b)-F(a)=\bm{\int}_a^b F'(x) \,dx.$

Linearity of integrals

$\bm{\int}_a^b (f+g) \,dx = \bm{\int}_a^b f \,dx + \bm{\int}_a^b g \,dx$ and $\bm{\int}_a^b c f \,dx = c \bm{\int}_a^b f \,dx.$

Additivity over intervals

If $a<c<b$, then $\bm{\int}_a^b f \,dx = \bm{\int}_a^c f \,dx + \bm{\int}_c^b f \,dx.$

Reversing bounds property

$$\bm{\int}_a^b f \,dx = -\bm{\int}_b^a f \,dx.$$

Zero-width interval property

$\int_a^a f(x) dx = 0$.

Even function

A function with $f(-x) = f(x)$; on $[-a,a]$, $\int_{-a}^a f(x) \,dx = 2\int_0^a f(x) \,dx.$

Odd function

A function with $f(-x) = -f(x)$; on $[-a,a]$, $\int_{-a}^a f(x) \,dx = 0.$

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 $\int_a^b f(x) dx \geq \int_a^b g(x) dx$.

Bounding (min/max) property

If m ≤ f(x) ≤ M on [a,b], then m(b−a) ≤ $\int_a^b f(x) dx$ ≤ 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)

$\bm{\int} x^n \,dx = \frac{x^{n+1}}{n+1} + C,$ for $n \neq -1$.

Logarithm antiderivative case

$\bm{\int} \, \frac{1}{x} \,dx = \ln|x| + C$ (the special case where $n=-1$).

Exponential antiderivative (e^x)

$$\int e^x \,dx = e^x + C.$$

Exponential antiderivative (a^x)

$$\int a^x \,dx = \frac{a^x}{\ln(a)} + C, \text{ for } a > 0 \text{ and } a \neq 1.$$

Basic trig antiderivative pair: cos

∫ cos x dx = sin x + C.

Basic trig antiderivative pair: sin

$$\bm{\int} \,\sin \,x \,dx = -\cos \,x + C.$$

u-substitution (integration by substitution)

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

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 $A(x)=\bm{\int}_a^x r(t) \,dt$, 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)+$\int_a^b r(t) dt$ (final amount = initial + change).