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Master Riemann Sum: Unlock the Power of Area Approximation

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

auto-gen

Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Parent Assignments

Challenge Zone

Foundation Review

math

intro

Ex. 1

Finding area under x squared from 1 to 3 using a graphing calculator

Ex. 2

E. 3

Ex. 4

Estimating area under x squared using trapezoid approximation

Understanding area under a curve and the idea behind Riemann sums

Understanding sigma notation and summation formulas

Understanding Riemann sum formulas and right, left, and middle methods

Expressing integrals as Riemann sums and computing exact area under x squared

Right Riemann sum for x squared on [1, 3] with 4 rectangles

Left Riemann sum for f(x)=x² on [1,3] with 4 subintervals

Computing a middle Riemann sum for x squared on [1, 3] with 4 subintervals

Right Riemann sum of x² - 5x + 3 on [2,5] with 6 subintervals

Evaluating a definite integral using the Riemann sum with infinite subintervals

In this lesson, we will learn how to approximate the area under the curve using rectangles.  First we notice that finding the area under the curve is easy if the function is a straight line. However, what if it is a curve? We will actually have to approximate curves using a method called "Riemann Sum". This method involves finding the length of each sub-interval (delta x), and finding the points of interest, finding the y values of each point of interest, and then use the find the area of each rectangle to sum them up. There are 3 methods in using the Riemann Sum. First is the "Right Riemann Sum", second is the "Left Riemann Sum", and third is the "Middle Riemann Sum". Lastly, we will look at the idea of infinite sub-intervals (which leads to integrals) to exactly calculate the area under the curve.

What is a Riemann sum?

Riemann Sum: The Key to Mastering Area Under the Curve

What You'll Learn

Understand the concept of approximating area under a curve using rectangles

Calculate delta x (width of rectangles) and identify points of interest in an interval

Apply the Right, Left, and Middle Riemann Sum methods to estimate areas

Use summation formulas to evaluate Riemann sums with infinite subintervals

Express Riemann sums as definite integrals when n approaches infinity

Recognize how increasing subintervals improves approximation accuracy

What You'll Practice

Computing Riemann sums using right, left, and middle endpoints

Approximating areas with a specified number of rectangles or trapezoids

Evaluating limits of Riemann sums as n approaches infinity

Using graphing calculators to verify area calculations

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

11 Video lessons

Practice exercises

Learning resources

Skills

Riemann Sum

Area Under Curve

Definite Integrals

Summation Formulas

Limits

Delta x

Approximation Methods

Calculus Foundations