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Browse materialsMIT 18.02SC · 3. Double Integrals and Line Integrals in the Plane

Multivariable Calculus — 3. Double Integrals and Line Integrals in the Plane

Subjects: Algebra · Calculus · Linear Algebra · Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

About this course

Source: MIT OpenCourseWare 18.02SC Multivariable Calculus, Fall 2010, Prof. Denis Auroux — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: this unit's sessions in MIT's listing order. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 22, 2026. How RedLectern is built.

Instructor: Prof. Denis Auroux
Term: Fall 2010
Level: Undergraduate
What's inside: 49 videos · 5 problem sets · 2 exams · 32 solutions · 52 other files
Materials open: Videos here · files on MIT OCW

Other versions of this course

18.02SC · Fall 2010 · Study here (this version)
18.02 · Fall 2007 · Study here
18.02 · Spring 2006 · Study here

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Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Session 47: Definition of Double Integration

8 files

Session 48: Examples of Double Integration

11 files

Session 49: Exchanging the Order of Integration

Session 50: Double Integrals in Polar Coordinates

8 files

Session 51: Applications: Mass and Average Value

7 files

Session 52: Applications: Moment of Inertia

11 files

Session 53: Change of Variables

13 files

Session 54: Example: Polar Coordinates

Session 55: Example

7 files

Problem Set 7

Session 56: Vector Fields

7 files

Session 57: Work and Line Integrals

13 files

Session 58: Geometric Approach

7 files

Session 59: Example: Line Integrals for Work

10 files

Session 60: Fundamental Theorem for Line Integrals

14 files

Session 61: Conservative Fields, Path Independence, Exact Differentials

Session 62: Gradient Fields

10 files

Session 63: Potential Functions

13 files

Session 64: Curl

9 files

Problem Set 8

Session 65: Green's Theorem

11 files

Session 66: Curl(F) = 0 Implies Conservative

7 files

Session 67: Proof of Green's Theorem

9 files

Session 68: Planimeter: Green's Theorem and Area

Session 69: Flux in 2D

15 files

Session 70: Normal Form of Green's Theorem

11 files

Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces

7 files

Session 72: Simply Connected Regions and Conservative Fields

Problem Set 9

Practice Exam

Session 73: Exam Review

12 files

Multivariable Calculus — 3. Double Integrals and Line Integrals in the Plane

Materials by section

Session 47: Definition of Double Integration

Session 48: Examples of Double Integration

Session 49: Exchanging the Order of Integration

Session 50: Double Integrals in Polar Coordinates

Session 51: Applications: Mass and Average Value

Session 52: Applications: Moment of Inertia

Session 53: Change of Variables

Session 54: Example: Polar Coordinates

Session 55: Example

Problem Set 7

Session 56: Vector Fields

Session 57: Work and Line Integrals

Session 58: Geometric Approach

Session 59: Example: Line Integrals for Work

Session 60: Fundamental Theorem for Line Integrals

Session 61: Conservative Fields, Path Independence, Exact Differentials

Session 62: Gradient Fields

Session 63: Potential Functions

Session 64: Curl

Problem Set 8

Session 65: Green's Theorem

Session 66: Curl(F) = 0 Implies Conservative

Session 67: Proof of Green's Theorem

Session 68: Planimeter: Green's Theorem and Area

Session 69: Flux in 2D

Session 70: Normal Form of Green's Theorem

Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces

Session 72: Simply Connected Regions and Conservative Fields

Problem Set 9

Practice Exam

Session 73: Exam Review

Exam

Session 47 Clip: Definition of Double Integration

Session 48 Clip: Examples of Double Integration

Session 48: Regions of Integration

Session 49 Clip: Exchanging the Order of Integration

Session 49: Changing the Order of Integration

Session 50 Clip: Double Integrals in Polar Coordinates

Session 50: Integration in Polar Coordinates

Session 51 Clip: Applications: Mass and Average Value

Session 52 Clip: Applications: Moment of Inertia

Session 52: Integrals with Density

Session 53 Clip: Change of Variables

Session 53: Change of Variables

Session 54 Clip: Polar Coordinates

Session 54: Integral of exp(-x²)

Session 55 Clip: Example

Session 55: Change of Variables

Session 56 Clip: Vector Fields

Session 57 Clip: Work and Line Integrals

Session 58 Clip: Geometric Approach

Session 58: Line Integrals by Geometric Reasoning

Session 59 Clip: Line Integrals

Session 60 Clip: Fundamental Theorem

Session 60: Fundamental Theorem of Line Integrals

Session 60 Clip: An Important Non-Conservative Field

Session 61 Clip: Conservative Fields, Path Independence, Exact

Session 62 Clip: Gradient Fields

Session 62: Non-Conservative Vector Fields

Session 63 Clip: Potential Functions

Session 63: Potentials of Gradient Fields

Session 64 Clip: Curl

Session 65 Clip: Green's Theorem

Session 65: Green's Theorem: An Off Center Circle

Session 66 Clip: Curl(F)=0 Implies Conservative

Session 67 Clip: Proof of Green's Theorem

Session 68 Clip: Planimeter: Green's Theorem and Area

Session 68: Application of Green's Theorem

Session 69 Clip: Flux Across a Curve

Session 69 Clip: Calculating Flux

Session 69: Flux Across a Curve