Accumulation and Riemann Sums: Free Course

About this course

Accumulation can be approximated by adding many small contributions. This course develops Riemann sums, summation notation, definite-integral properties and functions defined by integrals. Study the complete public source playlist from Khan Academy in English, with all 27 recordings in the on-site classroom. Original lesson titles, source order and repeated entries are preserved.

Course at a glance

What you’ll learn

• Build Riemann, midpoint and trapezoidal estimates.
• Express sums using sigma notation.
• Relate definite integrals to limits of sums.
• Use interval and linearity properties of integrals.

Skills you’ll gain

• Riemann sums
• Sigma notation
• Definite integrals
• Accumulation functions

Course Overview

Left, midpoint and trapezoidal approximations connect a graph or table with a numerical estimate. Over- and under-estimation examples use the shape of the function to reason about error direction. Sigma notation expresses a sum compactly, and limiting sums lead to the definition of a definite integral. Interval splitting, reversed bounds, scaling and sums of functions provide ways to manipulate integrals without immediately calculating an antiderivative. Accumulation-function examples make the upper or lower endpoint variable. Keep signed accumulation distinct from ordinary geometric area, especially when the graph lies below the axis or the interval direction is reversed.

How to study the course

• Videos 1–6: this part of the source list includes “Definite integrals intro” through “Breaking up integral interval”. Identify the function, variable and conditions. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 7–13: this part of the source list includes “Definite integral as the limit of a Riemann sum” through “Switching bounds of definite integral”. Connect the method to the meaning of the problem. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 14–20: this part of the source list includes “Definite integral properties (no graph): function combination” through “Worked example: Rewriting definite integral as limit of Riemann sum”. Rework a demonstration with a changed parameter. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 21–27: this part of the source list includes “Functions defined by integrals” through “Definite integral over a single point”. Check the result and explain its interpretation. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.

Who is this for?

This course suits learners with algebra and trigonometry preparation who want the specific calculus topic in the title. It also provides a recorded reference for revisiting a topic after an earlier lesson or study attempt. Choose examples appropriate to your background and allow time to work through the method instead of judging understanding only by how familiar the terminology sounds.

Before you start

Comfort manipulating algebraic expressions and interpreting functions is important. Review the prerequisite calculus stated by a lesson before using an advanced technique.

Put the lessons into practice

Use a short table of function values to form two different integral estimates and state the sampling rule. Sketch a compatible graph and discuss possible over- or under-estimation. Rewrite a definite integral as a limiting sum, then use interval splitting and reversed bounds to obtain related integrals without repeating the calculation.

This optional study activity is provided by WikiFree for independent practice. Keep your working notes and use the relevant classroom recording to check a step, definition or interpretation. It is not an instructor-marked assignment.

Certificate / Assessment

Watching the public videos and using WikiFree completion tracking does not award a certificate, academic credit, an examination voucher or instructor assessment. Use your practice output to review what you can explain and repeat independently. The recorded lessons provide study material; completion tracking records your progress through that material.

Frequently asked questions

Provider: Khan Academy. Videos play from the Khan Academy channel on YouTube.