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: $\int_a^b v(t) \, dt$.

Total distance traveled

Distance regardless of direction; computed as $\int_a^b |v(t)| \, dt$.

Net change theorem (FTC interpretation)

Relationship $Q(b) - Q(a) = \int_a^b Q'(t) \, dt$; net change equals the integral of the rate.

Accumulation function

A function defined by accumulating a rate from a fixed start: $A(x) = \int_a^x f(t) \, dt$.

Derivative of an accumulation function

If $A(x) = \int_a^x f(t) \, dt$, 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: $\Sigma f(sample point)Δx$.

Partition

A division of $[a,b]$ into subintervals $a=x_0<x_1<\ldots<x_n=b$ 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: $Ln = \sum_{i=1}^n f(x_{i-1}) \, \triangle x$.

Right Riemann sum (Rn)

Riemann sum using right endpoints $x_i$ as sample points: $Rn = \sum_{i=1}^n f(x_i) \, \triangle x$.

Midpoint Riemann sum (Mn)

Riemann sum using midpoints $m_i = \frac{x_{i-1} + x_i}{2}$ as sample points: $Mn = \sum f(m_i) \, \triangle x$.

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 $\Sigma$, such as $\sum_{i=1}^n a_i = a_1 + … + a_n$.

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 $\int_a^b f(x) \, dx$: 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).