Understanding Accumulation, Riemann Sums, and the Definite Integral (AP Calculus AB Unit 6)

Accumulation (in calculus)

The process of adding up many small contributions from a continuously varying rate of change to find a total (e.g., total change over an interval).

Rate of change

A quantity that describes how fast something changes with respect to an input (e.g., velocity in meters per second, flow in gallons per minute).

Accumulated change (total over an interval)

The overall amount a quantity changes over an interval, found by combining many small changes (often via an integral).

Unit analysis (rate × width)

Checking units to interpret accumulation: multiplying a rate (units per input) by a small input width cancels the input unit and yields units of the accumulated quantity.

Signed area

Area interpretation that counts regions above the axis as positive and regions below the axis as negative, matching net accumulation.

Net change

The overall change on an interval that combines positive and negative contributions (equivalent to signed area under a rate curve).

Total change (total variation)

The amount of change ignoring sign, found by adding magnitudes (often requires integrating an absolute value or splitting into sign-consistent intervals).

Area under a rate curve (interpretation)

For a rate function r(x), the accumulation from a to b is represented by the signed area between r and the x-axis on [a,b].

Accumulation function

A function defined by accumulating a rate up to a variable endpoint, typically A(x)=∫_a^x f(t) dt.

Dummy variable

The variable used inside an integral (e.g., t in ∫_a^x f(t)dt) that serves only as a placeholder and does not affect the outside variable.

Initial value of an accumulation function

A(a)=∫_a^a f(t) dt = 0; accumulating over an interval of zero length gives zero.

Displacement

Net change in position, commonly given by integrating velocity: s(t)=∫_0^t v(u) du (can be positive or negative).

Distance traveled (via integrals)

Total path length that ignores direction; often computed as ∫ |v(t)| dt or by splitting into intervals where velocity keeps a constant sign.

Riemann sum

An approximation of an integral/accumulation formed by summing rectangle areas f(x_i*)Δx over subintervals of [a,b].

Subinterval

One of the smaller intervals formed when [a,b] is partitioned into n pieces for a Riemann sum.

Equal subinterval width (Δx)

For n equal pieces of [a,b], the common width is Δx=(b−a)/n.

Partition points

The endpoints that divide [a,b] into subintervals, typically x_i = a + iΔx for i=0,1,…,n.

Sample point (x_i*)

A chosen x-value within the i-th subinterval used to set the rectangle height in a Riemann sum.

Left Riemann sum

A Riemann sum using left endpoints as sample points (typically xi* = x{i−1}).

Right Riemann sum

A Riemann sum using right endpoints as sample points (typically xi* = xi).

Midpoint Riemann sum

A Riemann sum using midpoints as sample points (xi* = (x{i−1}+x_i)/2 = a+(i−1/2)Δx).

Overestimate/underestimate rule (monotonic case)

If f is increasing on [a,b], left sums tend to underestimate and right sums tend to overestimate; if f is decreasing, the opposite tends to occur.

Summation notation (sigma notation)

A compact way to write repeated addition, such as a Riemann sum: ∑{i=1}^n f(xi*)Δx.

Definite integral

The limit of Riemann sums as the partition gets infinitely fine: ∫a^b f(x) dx = lim{n→∞} ∑{i=1}^n f(xi*)Δx (if the limit exists).

“Rate times width” structure

A setup check for accumulation: each small contribution should look like (rate)×(small input width), i.e., f(x_i*)Δx or ∫ f(x) dx.