Unit 6 Foundations: Accumulation, Riemann Sums, and Definite Integrals

Accumulation

The result of adding many small contributions over an interval (time, distance, etc.); in calculus, modeled with integrals.

Rate of change

A quantity describing how fast something changes with respect to an input variable (e.g., liters/min, m/s); often represented by a derivative.

Derivative

A measure of instantaneous rate of change; answers “how fast is it changing right now?”

Integral (definite integral)

Measures net accumulation of a function over an interval; answers “how much change built up over an interval?”

Net change

Overall change in a quantity over an interval, accounting for increases and decreases (negative contributions subtract).

Total change / Total accumulation

Amount accumulated without regard to direction; often requires absolute values (e.g., total distance).

Signed area

Interpretation of a definite integral where area above the x-axis counts positive and area below counts negative.

Displacement

Net change in position; computed as the integral of velocity: ∫_a^b v(t) dt.

Total distance traveled

Distance regardless of direction; computed as ∫_a^b |v(t)| dt.

Net change theorem (FTC interpretation)

Relationship Q(b) − Q(a) = ∫_a^b Q′(t) dt; net change equals the integral of the rate.

Accumulation function

A function defined by accumulating a rate from a fixed start: A(x)=∫_a^x f(t) dt.

Derivative of an accumulation function

If A(x)=∫_a^x f(t) dt, then A′(x)=f(x).

Units of an integral

Integral units equal (units of the integrand) × (units of the variable); e.g., (m/s)×(s)=m.

Riemann sum

An approximation of a definite integral by summing rectangle areas: Σ f(sample point)Δx.

Partition

A division of [a,b] into subintervals a=x0<x1<…<xn=b used for Riemann sums.

Subinterval width (Δx)

For an equal-width partition, Δx=(b−a)/n; the width of each rectangle in a Riemann sum.

Sample point (x*i)

A chosen point in each subinterval where the function is evaluated to form a rectangle height in a Riemann sum.

Left Riemann sum (Ln)

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

Right Riemann sum (Rn)

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

Midpoint Riemann sum (Mn)

Riemann sum using midpoints mi=(x{i−1}+xi)/2 as sample points: Mn=Σ f(mi)Δx.

Left/right sum bias for increasing functions

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

Left/right sum bias for decreasing functions

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

Summation (sigma) notation

Compact notation for adding many terms, written with Σ, such as Σ{i=1}^n ai = a1+…+an.

Index (in sigma notation)

The variable (often i) that counts terms in a sum; it runs from the lower limit to the upper limit.

Definite integral notation pieces

In ∫_a^b f(x) dx: a and b are bounds (start/end), f(x) is the rate/height, and dx indicates accumulation with respect to x (and supports unit interpretation).