Definite Integral - Limit of Riemann Sums


This section contains documents created from scanned original files and other
documents that could not be made accessible to screen reader software. A "#"
symbol is used to denote such documents.

Lecture Notes

The Problem of Areas

Document PDF
Section 1, Page 1 to page 2

Riemann integrals are introduced as a concept using the example of finding the area of a circle from the areas of N-sided polygons inscribed in the circle. Signed area (positive above the x-axis, negative below) is introduced.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None
Back to Top

Partitions

Document PDF
Section 2, Page 2

Interval partitions are defined, including the concepts of mesh size and fine vs. coarse partitions.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None
Back to Top

Riemann Sums

Document PDF
Section 3, Page 2 to page 3

Definition, including a discussion of partition choices when computing these sums.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Partitions (section 2 of this lecture)
Back to Top

The Riemann Integral

Document PDF
Section 4, Page 3 to page 4

Definite integrals are defined. Includes an example using the function f(x) = x.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Partitions and Riemann Sums (sections 2 and 3 of this lecture)
Back to Top

The Riemann Sum for the Exponential Function

Document PDF
Section 1, Page 1 to page 2

Using Riemann sums to find the Riemann integral for the function f(x) = ex.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (lecture 14)
Back to Top

The Riemann Sum for xr

Document PDF
Section 2, Page 3 to page 5

Using Riemann sums to find the Riemann integral for the function f(x) = xr.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (lecture 14)
Back to Top

Online Textbooks

Area and Notation

Document Document

Definition of the definite integral as the area under a curve, including definition of the integrand.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None
Back to Top

Riemann Sums

Document Document

Definition, including the use of Riemann sums in finding the area under a curve.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Area Under a Curve (OT20.1)
Back to Top

Practice Problems

Evaluating a Riemann Integral

Document PDF
Problem 2 (page 2)

Using upper sums to evaluate a definite integral.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Page 3

Back to Top

Exam Questions

Riemann Sums and Integrals

Document PDF
Problem 4.1 (page 3) to Problem 4.3 (page 3)

Three problems which involve evaluating Riemann sums and integrals.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF)

Back to Top

Limit Definition of Integral

Document PDF
Problem 2 (page 1)

Evaluating an integral using the definition of an integral as the limit of sums.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

Back to Top

Definite Integrals

Document PDF - 2.2 MB
Problem 3B-1 (page 21) to Problem 3B-7 (page 22)

Seven questions which involve using sigma notation for sums, computing Riemann sums for definite integrals, and evaluating limits by relating them to Riemann sums.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Page 39

Back to Top
Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound