Definite integral as the limit of a Riemann sum | AP Calculus AB | Khan Academy

1. Diagram
   Observation

   The graph is labeled y=f(x) in purple above the plotted curve.

2. Audio
   Observation

   The explanation selects the left endpoint to set the pictured rectangle height.

f(x)

The function whose graph bounds the area being approximated by rectangles.

A real-valued function on an interval [a,b]; the video does not state further regularity assumptions.

1. Diagram
   Observation

   The x-axis shows a yellow label a at the left end of the shaded region.

2. Audio
   Observation

   The explanation identifies the interval bounded by a and b.

a

Left endpoint of the interval over which the area is approximated.

A real number serving as the lower boundary of the partition interval.

1. Diagram
   Observation

   The x-axis shows a yellow label b at the right end of the shaded region.

2. Audio
   Observation

   The explanation identifies the interval bounded by a and b.

b

Right endpoint of the interval over which the area is approximated.

A real number serving as the upper boundary of the partition interval.

1. Diagram
   Observation

   The summation formula has upper index n, and the Δx formula divides by n.

2. Audio
   Observation

   The equal partition gives the common width of the rectangle bases.

n

Number of subintervals or rectangles used in the displayed approximation.

A positive integer; the video does not explicitly state this, but the formulas use it as a counting parameter.

1. Formula
   Observation

   The written formula states “where Δx = (b-a)/n”.

2. Audio
   Observation

   The equal partition gives the common width of the rectangle bases.

Δx

Width of each rectangle in the equal-partition example.

Defined in the displayed formula as (b-a)/n for the shown case.

1. Formula
   Observation

   The summation term is f(x_{i-1}).

2. Audio
   Observation

   The explanation selects the left endpoint to set the pictured rectangle height.

1. The video does not write a separate formula defining x_{i-1}; its meaning is inferred from the spoken left-endpoint description and the summation notation.

x_{i-1}

Sample point used for the i-th rectangle in the displayed left-endpoint sum; visually corresponds to the left endpoint of that subinterval.

A point in the i-th subinterval of [a,b] under the equal-partition setup.

1. Formula
   Observation

   The sigma notation is written with i=1 below Σ and n above Σ.

i

Summation index running from 1 to n over the rectangles.

Integer index in the displayed sum.

1. Formula
   Observation

   n appears as the upper limit of the summation and in Δx = (b-a)/n.

2. Audio
   Observation

   The lesson uses more and narrower rectangles to illustrate improving approximations.

n

Number of subintervals used in the Riemann sum.

Positive integer; the definition later takes n → ∞.

1. Formula
   Observation

   i appears in the summation index i=1 to n.

i

Summation index labeling one rectangle/subinterval.

1. Formula
   Observation

   x_{i-1} appears inside f(x_{i-1}) in the sum.

2. Diagram
   Observation

   The graph shows sample points labeled 1, 2, 3, ..., n under the curve.

1. No coordinate formula for the partition points is separately written; the displayed and earlier narrated sample is the left endpoint.

x_{i-1}

The left endpoint of the displayed subinterval; the earlier section specifies this sampling choice.

1. Formula
   Observation

   Δx appears multiplied by f(x_{i-1}) in the sum.

2. Formula
   Observation

   The formula states Δx = (b-a)/n.

Δx

Width of each subinterval in the displayed Riemann sum.

1. Formula
   Observation

   a appears as the lower endpoint in Δx = (b-a)/n.

2. Diagram
   Observation

   a labels the left endpoint of the interval on the x-axis.

3. Audio
   Observation

   The explanation identifies the interval bounded by a and b.

a

Left endpoint of the integration interval.

1. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

2. Formula
   Observation

   The board shows Σ_{i=1}^{n} f(x_{i-1}) Δx, where Δx = (b-a)/n.

3. Diagram
   Observation

   The speaker writes “Riemann Sum” and draws an arrow toward the displayed summation formula.

1. The clip gives a contextual explanation rather than a fully formal definition of a general Riemann sum.

Left, right and midpoint sums choose one sample value in each subinterval. Unequal widths are also allowed. The lesson additionally mentions the trapezoidal rule; this related quadrature uses the average of endpoint values, rather than the standard single-tag rectangle formula.

1. The example shown uses an interval [a,b].

2. The displayed formula uses equal subinterval widths.

3. The sample points shown correspond to left endpoints.

1. Approximating area by summing rectangle areas

1. Audio
   Observation

   The explanation selects the left endpoint to set the pictured rectangle height.

2. Diagram
   Observation

   The blue rectangles under y=f(x) have heights determined by values on the curve at their left edges.

3. Formula
   Observation

   The summation uses f(x_{i-1}) as the sampled height.

For the specific picture on the board, each rectangle’s height is obtained by evaluating f at the left endpoint of its subinterval. Multiplying those heights by the common width Δx and summing gives the displayed approximation to the area under the curve.

1. The interval [a,b] is partitioned into n equal subintervals.

2. Each rectangle uses the left endpoint of its subinterval as the sample point.

1. Equal-width partition formula

1. Audio
   Observation

   The equal partition gives the common width of the rectangle bases.

2. Formula
   Observation

   The board writes Δx = (b-a)/n.

When [a,b] is divided into n equal subintervals, each subinterval has width Δx=(b-a)/n. This is the width used in the displayed rectangle sum.

1. The partition is uniform.

2. There are n subintervals between a and b.

1. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

2. Formula
   Observation

   The board shows a summation over rectangle contributions.

The clip reviews the basic strategy of estimating the area under a curve by decomposing the region into rectangles and adding their areas. The displayed sigma notation compactly represents that total.

1. The region is bounded above by y=f(x) and below by the x-axis on [a,b] in the pictured example.

1. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

2. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

1. The clip names these variants verbally but does not display formulas for right-endpoint, midpoint, trapezoidal, or unequal-partition sums.

Left, right and midpoint sums choose one sample value in each subinterval. Unequal widths are also allowed. The lesson additionally mentions the trapezoidal rule; this related quadrature uses the average of endpoint values, rather than the standard single-tag rectangle formula.

1. The statement is about the broader class of constructions called Riemann sums.

2. The displayed formula is only one special case.

1. Riemann sum as the general name for the displayed area approximation

1. Formula
   Observation

   Σ_{i=1}^n f(x_{i-1}) Δx, where Δx = (b-a)/n is shown on screen.

2. Diagram
   Observation

   Rectangles under y=f(x) on [a,b] are drawn to illustrate the sum.

3. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

1. The video does not state general conditions on f or on the choice of sample points x_{i-1}.

The clip displays a Riemann sum as the total of rectangle areas f(x_{i-1})Δx for i=1,...,n, with equal subinterval width Δx=(b-a)/n. The speaker emphasizes that this is one example of a Riemann sum, not the only possible one.

1. Interval endpoints are a and b.

2. There are n subintervals.

3. The displayed version uses equal width Δx=(b-a)/n.

1. n
2. i
3. x_{i-1}
4. Δx
5. a
6. b
7. f(x)
8. Σ

1. Formula
   Observation

   lim_{n→∞} Σ_{i=1}^n f(x_{i-1}) Δx is written before the sum.

2. Formula
   Observation

   ∫_a^b f(x) dx is written as the resulting notation.

3. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. The video does not explicitly state existence conditions for the limit or integrability assumptions on f.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Use a Riemann sum over [a,b].

2. Take the limit as n→∞.

3. The result is denoted by the definite integral from a to b.

4. Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Riemann sum formula shown on screen
2. lim_{n→∞}
3. ∫_a^b f(x) dx
4. dx

1. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

Left, right and midpoint sums choose one sample value in each subinterval. Unequal widths are also allowed. The lesson additionally mentions the trapezoidal rule; this related quadrature uses the average of endpoint values, rather than the standard single-tag rectangle formula.

1. Applies when approximating definite integrals using finite sums.

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

2. Formula
   Observation

   The equation $\lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i-1}) \Delta x = \int_{a}^{b} f(x) dx$ is displayed.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. The limit must exist.

2. n represents the number of subintervals.

3. $\Delta x$ represents the width of the subintervals.

4. Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Types of Riemann Sums

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. At this early item’s time the next part has not yet been developed; it is continued later in the same complete video.

The speaker states that the Riemann sum is used to define the Riemann integral.

1. The context is a first-year calculus course.

2. The discussion concerns the standard role of Riemann sums in integration theory.

No explicit quantifier is stated in the clip.

1. Audio
   Observation

   The portrait is identified as Riemann, linking his name to the terminology.

2. Diagram
   Observation

   A portrait is shown with the handwritten label “Bernhard Riemann”.

The clip attributes the name “Riemann sums” to Bernhard Riemann.

Universal naming attribution as stated by the speaker; no formal logical quantifier is given.

1. Audio
   Observation

   The lecturer introduces the Riemann approach as a formal definition used in calculus.

1. This is presented as the lecturer's characterization rather than a proved statement within the clip.

The Riemann integral is presented as a mainstream formal or rigorous definition of the integral.

No explicit quantifier is stated in the clip.

1. Audio
   Observation

   The lesson uses more and narrower rectangles to illustrate improving approximations.

2. Diagram
   Observation

   The initial drawing shows finitely many rectangles approximating the area under the curve.

1. The clip does not prove this monotonic improvement claim; it is stated verbally.

Refinement gives convergence under the integrability and shrinking-mesh conditions. This does not guarantee that every increase in the number of rectangles strictly reduces the absolute error.

1. A Riemann sum is being used to approximate the area under a curve on [a,b].

2. Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

For Riemann-integrable f and tagged partitions whose mesh tends to zero; no stepwise strict-error comparison asserted.

1. Audio
   Observation

   The explanation says the limiting idea is not restricted to the one drawn sampling choice.

1. The clip does not specify the precise class of allowed Riemann sums or prove independence of the choice.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

All valid tags and shrinking-mesh partitions for a Riemann-integrable function.

1. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

2. Formula
   Observation

   The board displays Σ_{i=1}^{n} f(x_{i-1}) Δx with Δx=(b-a)/n.

3. Diagram
   Observation

   The rectangles under y=f(x) visually match a left-endpoint construction.

1. The clip does not write out the intermediate algebraic derivation step by step; it presents the final sigma form directly.

1. Expression

   $$$$
   Explanation

   Start with the geometric picture of several rectangles approximating the area under y=f(x) on [a,b].

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Shown in the video
2. Expression

   $$\Delta x=\frac{b-a}{n}$$
   Explanation

   Because the interval is equally partitioned into n pieces, each rectangle has the same width.

   Justification

   Explicitly stated in the audio and written on the board.

   Shown in the video
3. Expression

   $$f(x_{i-1})$$
   Explanation

   For the i-th rectangle, the height is taken from the function value at the left endpoint of that subinterval.

   Justification

   Explicitly stated in the audio and consistent with the diagram.

   Shown in the video
4. Expression

   $$\sum_{i=1}^{n} f(x_{i-1})\Delta x$$
   Explanation

   Adding the areas of all n rectangles gives the displayed finite sum.

   Justification

   Area of each rectangle is height times width; summing over i yields the sigma expression.

   Derived from the video

The pictured left-endpoint equal-width rectangle approximation is summarized by the formula Σ_{i=1}^{n} f(x_{i-1})Δx with Δx=(b-a)/n.

1. Formula
   Observation

   lim_{n→∞} is added in front of the Riemann sum.

2. Animation
   Observation

   A second diagram is drawn with many more narrow rectangles under the curve.

3. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

4. Formula
   Observation

   The final notation ∫_a^b f(x) dx is written.

1. The derivation is conceptual and visual; the clip does not provide an epsilon-delta proof or discuss convergence criteria.

1. Expression

   $$\sum_{i=1}^{n} f(x_{i-1})\,\Delta x$$
   Explanation

   Start with a finite Riemann sum representing the total area of n rectangles under the curve.

   Justification

   This is the formula already written on screen and identified verbally as a Riemann sum.

   Shown in the video
2. Expression

   $$\lim_{n\to\infty}\sum_{i=1}^{n} f(x_{i-1})\,\Delta x$$
   Explanation

   For the equal partition, increasing n makes all subinterval widths tend to zero; with Riemann integrability the sums converge.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Supplementary explanation
3. Expression

   $$\int_a^b f(x)\,dx$$
   Explanation

   Denote the common limit by the definite integral; the positive example permits an ordinary-area interpretation.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Supplementary explanation

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Audio
   Observation

   The explanation links shrinking finite widths to the dx notation and later cautions that its infinitesimal picture is intuition.

2. Diagram
   Observation

   Visual annotation 'infinitely small' added to dx.

1. Speaker notes this is not a rigorous way of thinking about it.

1. Expression

   $$\Delta x$$
   Explanation

   Start with the finite width of a rectangle in a Riemann sum.

   Justification

   Definition of Riemann sum components.

   Shown in the video
2. Expression

   $$\Delta x \to 0$$
   Explanation

   Consider the width becoming infinitely small.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Shown in the video
3. Expression

   $$dx$$
   Explanation

   The author uses an infinitely-small-width picture and explicitly calls it non-rigorous. In the usual real-variable Riemann integral, dx marks the integration variable; it is not a finite rectangle width, a newly defined nonzero infinitesimal real number, or the real number zero. The definition uses limits of finite sums.

   Justification

   Notation for the differential.

   Supplementary explanation

The author uses an infinitely-small-width picture and explicitly calls it non-rigorous. In the usual real-variable Riemann integral, dx marks the integration variable; it is not a finite rectangle width, a newly defined nonzero infinitesimal real number, or the real number zero. The definition uses limits of finite sums.

1. Diagram
   Observation

   A graph of y=f(x) over [a,b] is shown with several blue rectangles beneath the curve.

2. Formula
   Observation

   The board writes Σ_{i=1}^{n} f(x_{i-1}) Δx, where Δx = (b-a)/n.

3. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

1. No numerical function, interval endpoints, or value of n are specified, so no numeric answer can be computed from the clip.

Approximate the area under y=f(x) from x=a to x=b using equally spaced rectangles whose heights come from left endpoints.

1. Function graph y=f(x).

2. Interval endpoints a and b.

3. n equal subintervals.

4. Rectangle heights use left endpoints x_{i-1}.

Express the rectangle-area approximation in sigma notation and identify it as a Riemann sum.

1. Expression

   $$\Delta x=\frac{b-a}{n}$$
   Explanation

   Partition [a,b] into n equal subintervals, giving each rectangle the same width.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Shown in the video
2. Expression

   $$\text{height}_i=f(x_{i-1})$$
   Explanation

   Use the function value at the left endpoint of the i-th subinterval as the rectangle height.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Shown in the video
3. Expression

   $$\sum_{i=1}^{n} f(x_{i-1})\Delta x$$
   Explanation

   Sum the areas of all rectangles to obtain the approximation.

   Justification

   Follows from adding height times width over all subintervals.

   Shown in the video
4. Expression

   $$\text{Riemann Sum}$$
   Explanation

   Label the resulting finite sum as a Riemann sum.

   Justification

   Uses the rectangle-width and sampling construction explained in the lesson.

   Shown in the video

The displayed approximation is the Riemann sum $\sum_{i=1}^{n} f(x_{i-1})\Delta x$ with $\Delta x=\frac{b-a}{n}$.

The formula matches the visual construction: equal-width rectangles under y=f(x) on [a,b] with heights taken at left endpoints.

1. Diagram
   Observation

   The screen shows a portrait labeled “Bernhard Riemann” on the left, a coordinate graph with y=f(x) and rectangles on the upper right, and a summation formula below.

2. Animation
   Observation

   A cursor moves among the portrait, the rectangles, the labels a and b, and the formula.

1. Portrait labeled “Bernhard Riemann”.

2. Coordinate axes x and y.

3. Curve labeled y=f(x).

4. Blue rectangles under the curve on [a,b].

5. Formula Σ_{i=1}^{n} f(x_{i-1}) Δx, where Δx=(b-a)/n.

1. The cursor points first to the rectangles and interval endpoints.

2. It then moves to the summation formula.

3. Later it points to the portrait and the handwritten label “Riemann Sum”.

1. The graph remains on [a,b] throughout.

2. The displayed formula remains the same throughout the clip.

3. The rectangles continue to represent a left-endpoint equal-width construction.

The visual arrangement links the historical figure, the geometric rectangle picture, and the symbolic Riemann-sum formula as three representations of the same idea.

1. Animation
   Observation

   Handwritten text appears near the formula.

2. Diagram
   Observation

   The new label reads “Riemann Sum” with an arrow pointing toward the summation expression.

1. Exact stroke-by-stroke timing of the handwriting is approximate from the sampled frames.

1. Handwritten words “Riemann Sum”.

2. Arrow pointing to the summation formula.

1. The phrase “Riemann Sum” is written on the board.

2. An arrow is drawn from the phrase to the formula.

1. The underlying formula Σ_{i=1}^{n} f(x_{i-1}) Δx does not change.

This annotation explicitly identifies the displayed finite sum as an instance of a Riemann sum.

1. Diagram
   Observation

   A portrait labeled 'Bernhard Riemann' appears at left; a graph of y=f(x) with rectangles on [a,b] appears center; the formula Σ_{i=1}^n f(x_{i-1})Δx, where Δx=(b-a)/n, is written below; 'Riemann Sum' is written at right with an arrow.

1. Portrait labeled Bernhard Riemann

2. Coordinate axes x and y

3. Curve labeled y=f(x)

4. Rectangles under the curve on [a,b]

5. Formula Σ_{i=1}^n f(x_{i-1})Δx

6. Definition Δx=(b-a)/n

7. Label 'Riemann Sum'

1. The cursor points among the summation symbol, the rectangles, and the formula components.

1. The interval endpoints a and b remain fixed.

2. The curve y=f(x) remains the same in the first diagram.

The board visually links the geometric rectangle approximation to the algebraic Riemann-sum formula.

1. Animation
   Observation

   A new coordinate system is drawn at lower left, then a curve and many narrow blue rectangles between a and b.

2. Audio
   Observation

   The lesson uses more and narrower rectangles to illustrate improving approximations.

1. The exact number of rectangles is not specified; the drawing is schematic.

1. New x-y axes

2. Curve on [a,b]

3. Many narrow blue rectangles

4. Labels a and b

1. A second diagram is added below the original content.

2. The rectangles are much thinner and more numerous than in the first diagram.

1. The conceptual interval remains from a to b.

2. The purpose remains approximating area under the curve.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Formula
   Observation

   ∫_a^b f(x) dx is written to the right of the limit expression.

2. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. Expression lim_{n→∞} Σ_{i=1}^n f(x_{i-1})Δx

2. New notation ∫_a^b f(x) dx

1. The integral notation is added after the limit expression is established.

1. The same interval [a,b] and function f are referenced.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Animation
   Observation

   The words 'infinitely small' are handwritten next to the dx term in the integral formula.

1. Text 'infinitely small'

2. Formula term dx

1. Text appears on screen.

1. The rest of the formula remains unchanged.

The author uses an infinitely-small-width picture and explicitly calls it non-rigorous. In the usual real-variable Riemann integral, dx marks the integration variable; it is not a finite rectangle width, a newly defined nonzero infinitesimal real number, or the real number zero. The definition uses limits of finite sums.

1. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

2. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

One might think a Riemann sum must always use equal-width rectangles with left-endpoint heights.

The clip states that the displayed construction is only one particular instance; more general Riemann sums allow different sample points and unequal partitions.

1. Audio
   Observation

   The explanation says the limiting idea is not restricted to the one drawn sampling choice.

One might think the specific sum Σ_{i=1}^n f(x_{i-1})Δx with Δx=(b-a)/n is the only Riemann sum that matters.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Audio
   Observation

   The lesson contrasts a finite rectangle approximation with the limit represented by the integral.

2. Animation
   Observation

   Increasing n is shown to improve the approximation.

1. This misconception is inferred from the contrast made in the video rather than named directly.

A finite sum must always equal the integral, or conversely can never equal it.

Finite sums are approximations in general; special functions and sampling choices can give equality already. The limiting definition ensures the common integral under the integrability and shrinking-mesh conditions.

1. Audio
   Observation

   The explanation links shrinking finite widths to the dx notation and later cautions that its infinitesimal picture is intuition.

Thinking of dx simply as an 'infinitely small number' is mathematically rigorous.

The author uses an infinitely-small-width picture and explicitly calls it non-rigorous. In the usual real-variable Riemann integral, dx marks the integration variable; it is not a finite rectangle width, a newly defined nonzero infinitesimal real number, or the real number zero. The definition uses limits of finite sums.

1. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

2. Formula
   Observation

   The board shows the sigma expression corresponding to the rectangle picture.

The general method of summing rectangle areas is applied here through the specific left-endpoint rule shown on the board.

1. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

2. Diagram
   Observation

   The label “Riemann Sum” is attached to the specific formula.

The named concept “Riemann sum” generalizes the specific left-endpoint equal-width formula shown in the example.

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. At this early item’s time the next part has not yet been developed; it is continued later in the same complete video.

The first section announces the integral connection; the subsequent sections write the limit and continue the dx discussion.

1. Formula
   Observation

   lim_{n→∞} Σ_{i=1}^n f(x_{i-1})Δx = ∫_a^b f(x) dx is assembled across the board.

2. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Audio
   Observation

   The explanation links shrinking finite widths to the dx notation and later cautions that its infinitesimal picture is intuition.

2. Formula
   Observation

   Δx appears in the sum and dx appears in the integral notation.

1. At this early item’s time the next part has not yet been developed; it is continued later in the same complete video.

The width-to-notation comparison begins here and continues later in the complete video.

1. Audio
   Observation

   The explanation links shrinking finite widths to the dx notation and later cautions that its infinitesimal picture is intuition.

The author uses an infinitely-small-width picture and explicitly calls it non-rigorous. In the usual real-variable Riemann integral, dx marks the integration variable; it is not a finite rectangle width, a newly defined nonzero infinitesimal real number, or the real number zero. The definition uses limits of finite sums.

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

Editorial scope: f is Riemann integrable on the finite interval [a,b], with a<b; continuity is sufficient. Sample points belong to their subintervals and the largest subinterval width tends to zero. For the displayed equal partition, n→∞ ensures this. A nonnegative f gives ordinary area; a sign-changing f gives net signed area.

1. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

2. Diagram
   Observation

   The board writes “Riemann Sum” next to the formula.

1. Riemann sum as the general name for the displayed area approximation
2. Generality of Riemann sums beyond the displayed example

1. Audio
   Observation

   The explanation selects the left endpoint to set the pictured rectangle height.

2. Formula
   Observation

   The summation uses f(x_{i-1}).

1. Left-endpoint rectangle rule in the displayed example
2. x_{i-1}

1. Audio
   Observation

   The equal partition gives the common width of the rectangle bases.

2. Formula
   Observation

   The board writes Δx=(b-a)/n.

1. Equal-width partition formula

1. Audio
   Observation

   The lesson discusses other sampling choices and also mentions trapezoidal approximation.

1. Generality of Riemann sums beyond the displayed example
2. Mistaking the displayed example for the whole definition

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. At this early item’s time the next part has not yet been developed; it is continued later in the same complete video.

1. Riemann sums are used to define the Riemann integral
2. Riemann sum as the general name for the displayed area approximation

1. Formula
   Observation

   Σ_{i=1}^n f(x_{i-1})Δx, where Δx=(b-a)/n is shown.

2. Audio
   Observation

   The explanation connects the rectangle-area total with the displayed finite summation.

1. Riemann sum formula shown on screen

1. Formula
   Observation

   lim_{n→∞} Σ_{i=1}^n f(x_{i-1})Δx = ∫_a^b f(x) dx is written.

2. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. Definite integral defined as the limit of a Riemann sum
2. From finite Riemann sums to the definite integral notation

1. Audio
   Observation

   The lesson uses more and narrower rectangles to illustrate improving approximations.

2. Animation
   Observation

   A second diagram shows many more narrow rectangles.

1. Convergence as partitions become finer
2. Visualization of n→∞ using many narrow rectangles

1. Audio
   Observation

   The explanation says the limiting idea is not restricted to the one drawn sampling choice.

1. The limiting definition is not tied to one specific displayed sum
2. Mistaking the displayed formula for the only possible Riemann sum

1. Audio
   Observation

   The explanation links shrinking finite widths to the dx notation and later cautions that its infinitesimal picture is intuition.

1. dx
2. Conceptualizing the Differential dx

1. Audio
   Observation

   The explanation connects the limit of the finite sums to the definite-integral notation.

1. Riemann Definition of the Definite Integral