Foundations of Integration: From Accumulation to Riemann Sums

Integration

The accumulation of change over an interval.

Rate of change

The instant speed of change at a single point, often represented by a derivative.

Net change

The total amount of quantity accumulated, considering both increases and decreases.

Area under the curve

Represents the total change in quantity, such as displacement, on a graph.

Positive area

When the rate curve is above the axis, indicating an increase in quantity.

Negative area

When the rate curve is below the axis, indicating a decrease in quantity.

Net accumulation

The result of subtracting the area below the x-axis from the area above.

Riemann Sum

A method for approximating the area under a curve by dividing it into smaller shapes.

Left Riemann Sum (LRAM)

Uses the height from the left endpoint of each sub-interval for area approximation.

Right Riemann Sum (RRAM)

Uses the height from the right endpoint of each sub-interval for area approximation.

Midpoint Riemann Sum (MRAM)

Uses the height from the midpoint of each sub-interval for area approximation.

Overestimate

When a Riemann sum calculation results in a value greater than the actual area under the curve.

Underestimate

When a Riemann sum calculation results in a value less than the actual area under the curve.

Definite Integral

The limit of a Riemann sum as the number of rectangles approaches infinity, providing an exact area.

Sigma notation

A notation used to represent the sum of a sequence, often used in the context of Riemann sums.

Limit of a Riemann Sum

The process of taking the limit of a sum to define the exact area under a curve.

Concavity

The curvature of a function, which can affect the estimation error in area approximations.

Sub-intervals

Divisions of an interval used to simplify the calculation of area under a curve.

Interval width (Δx)

The width of each sub-interval, calculated as (b-a)/n.

Sample point (ck)

A point in each sub-interval at which the function is evaluated.

Increasing function

A function where f'(x) > 0, resulting in an underestimation by LRAM.

Decreasing function

A function where f'(x) < 0, resulting in an overestimation by LRAM.

Worked example

A specific problem used to illustrate the method for estimating quantities such as area.

Common mistakes

Frequent errors made when applying concepts of Riemann sums and integrals.

Notation mix-up

Confusion in recognizing the relationship between sums and integral notation.