Unit 6: Integration and Accumulation of Change

Derivative

A measure of instantaneous rate of change; answers “At this instant, how fast is something changing?”

Integral

A measure of accumulated change over an interval; adds up contributions of a varying rate across many small pieces.

Rate of change function

A function (often written r(t) or f(x)) that gives change per unit of the input (e.g., liters/min, m/s, dollars/day).

Accumulation

The total change gathered over an interval by adding up small amounts of “rate × small width,” leading to an integral.

Antiderivative

A function F whose derivative is the integrand: F'(x)=f(x); used to compute definite integrals via FTC.

Definite integral

An integral with bounds, ∫_a^b f(x)dx, representing net (signed) accumulation from x=a to x=b.

Indefinite integral

A family of antiderivatives written ∫ f(x)dx = F(x)+C (no bounds).

Constant of integration (+C)

The constant added to an antiderivative because differentiating any constant gives 0, so antidifferentiation cannot recover it.

Net change

Accumulated change that counts positive contributions and subtracts negative contributions (signed accumulation).

Total accumulated amount

A nonnegative total that often requires integrating an absolute value when the rate can be negative (e.g., total distance).

Displacement

Net change in position over an interval; computed by integrating velocity: ∫_a^b v(t)dt.

Speed

The magnitude of velocity, |v(t)|; used when computing total distance traveled.

Total distance traveled

Total path length; computed by integrating speed: ∫_a^b |v(t)|dt.

Net Change Theorem

If Q changes with rate Q'(t), then Q(b)−Q(a)=∫_a^b Q'(t)dt.

Initial value (in accumulation problems)

A starting amount (e.g., V(0)=50) that must be added to net change to get the final amount: V(b)=V(a)+∫_a^b r(t)dt.

Riemann sum

An approximation of an integral using rectangles: Σ f(x_i^*)Δx.

Partition

A division of an interval [a,b] into subintervals used to form Riemann or trapezoidal approximations.

Subinterval

One piece of a partition; has a width (often Δx or Δt) used as the base of a rectangle or trapezoid.

Δx (delta x)

The width of each equal subinterval in a partition: Δx=(b−a)/n.

Representative point (x_i^*)

A chosen sample point in a subinterval (left, right, midpoint, etc.) where the function value sets the rectangle height.

Left Riemann sum

A Riemann sum using left endpoints of subintervals for x_i^* (commonly excludes the far-right table value).

Right Riemann sum

A Riemann sum using right endpoints of subintervals for x_i^* (commonly excludes the far-left table value).

Midpoint Riemann sum

A Riemann sum using midpoints of subintervals for x_i^*; requires function values at midpoints.

Overestimate/underestimate rule (monotonicity)

If f is increasing, left sums tend to underestimate and right sums tend to overestimate; if f is decreasing, the roles reverse.

Definite integral as a limit

The definition ∫a^b f(x)dx = lim{n→∞} Σ f(x_i^*)Δx (when the limit exists).

Signed area

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

Bounds of integration

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

Integrand

The function being integrated, f(x), inside the integral ∫_a^b f(x)dx.

Differential (dx)

Indicates the variable of integration and the “width direction” (e.g., integrate with respect to x).

Reversing bounds property

∫a^b f(x)dx = −∫b^a f(x)dx.

Zero-width integral property

∫_a^a f(x)dx = 0.

Additivity across intervals

∫a^b f(x)dx = ∫a^c f(x)dx + ∫_c^b f(x)dx (with matching bounds).

Constant multiple rule (linearity)

∫a^b kf(x)dx = k∫a^b f(x)dx.

Sum rule for integrals

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

Comparison property

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

Fundamental Theorem of Calculus (FTC)

The bridge between derivatives and integrals: connects accumulation (integrals) to rates of change (derivatives) and antiderivatives.

FTC Part 1

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

Accumulation function

A function defined by a variable-limit integral, typically A(x)=∫_a^x f(t)dt, representing “total so far.”

Dummy variable

A placeholder variable inside an integral (like t in ∫ f(t)dt) that does not affect the outside variable.

Chain rule version of FTC (upper limit g(x))

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

FTC with both bounds varying

d/dx[∫_{h(x)}^{g(x)} f(t)dt] = f(g(x))g'(x) − f(h(x))h'(x).

FTC Part 2

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

Evaluate a definite integral using an antiderivative

Compute F(b)−F(a) after finding an antiderivative F of f (top bound minus bottom bound).

Power rule for integrals

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

Trig antiderivative

Reversing basic trig derivatives, e.g., since (sin x)'=cos x, then ∫ cos x dx = sin x + C.

U-substitution (substitution)

A method that rewrites an integral by letting u be an inner expression, converting the integral to one in u, then substituting back.

Trapezoidal Rule

A numerical method that approximates an integral by summing trapezoids formed by connecting endpoints of data on each subinterval.

Area of a trapezoid (for numerical integration)

(1/2)(b1+b2)h; in integrals, bases are endpoint heights and h is the subinterval width.

Trapezoidal error (concavity test)

If f is concave up, the trapezoidal rule tends to overestimate; if f is concave down, it tends to underestimate.

Concavity of an accumulation function

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