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README.md

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# <span style="color:rgb(213,80,0)">Multivariable Calculus: Integrals</span>
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Learn concepts and practice computing multiple integrals, line integrals, and surface integrals.
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[![View Multivariable: Integrals on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://www.mathworks.com/matlabcentral/fileexchange/181588-multivariable-integrals) or [![Open in MATLAB Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj&file=README.mlx)
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[![View on File Exchange](https://www.mathworks.com/matlabcentral/images/matlab-file-exchange.svg)](https://www.mathworks.com/matlabcentral/fileexchange/181588-multivariable-integrals) or [![Open in MATLAB Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj&file=README.mlx)
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[![MATLAB Versions Tested](https://img.shields.io/endpoint?url=https%3A%2F%2Fraw.githubusercontent.com%2FMathWorks-Teaching-Resources%2FMultivariable-Integrals%2Frelease%2FImages%2FTestedWith.json)](https://MathWorks-Teaching-Resources.github.io/Multivariable-Integrals)
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You can use these live scripts as demonstrations in lectures, class activities, or interactive assignments outside class. The first script released in this collection is It also includes examples of computing work and area.
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The instructions inside the live scripts will guide you through the exercises and activities. Get started with each live script by running it one section at a time. To stop running the script or a section midway (for example, when an animation is in progress), use the <img src="Images/EndIcon.png" width="19" alt="EndIcon.png"> Stop button in the **RUN** section of the **Live Editor** tab in the MATLAB Toolstrip.
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## Contact Us
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| **Courseware Module** <br> | **Sample content** <br> | **Available on:** <br> |
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| :-- | :-- | :-- |
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| [**Vector Arithmetic**](https://www.mathworks.com/matlabcentral/fileexchange/94555-vector-arithmetic) <br> | <img src="Images/image_1.png" width="171" alt="image_1.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/94555-vector-arithmetic) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Vector-Arithmetic&project=VectorArithmetic.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Vector-Arithmetic) <br> |
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| [**Multivariable: Space and Functions**](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> | <img src="Images/image_4.png" width="159" alt="image_4.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Space-and-Functions&project=Space.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Multivariable-Space-and-Functions) <br> |
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| [**Calculus: Integrals**](https://www.mathworks.com/matlabcentral/fileexchange/105740-calculus-integrals) <br> | <img src="Images/image_7.png" width="171" alt="image_7.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/105740-calculus-integrals) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Calculus-Integrals&project=Integrals.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Calculus-Integrals) <br> |
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| [**Vector Arithmetic**](https://www.mathworks.com/matlabcentral/fileexchange/94555-vector-arithmetic) <br> | <img src="Images/VectorArithmeticIcon.png" width="171" alt="VectorArithmeticIcon.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/94555-vector-arithmetic) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Vector-Arithmetic&project=VectorArithmetic.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Vector-Arithmetic) <br> |
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| [**Multivariable: Space and Functions**](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> | <img src="Images/MultivariableFunctionsIcon.png" width="159" alt="MultivariableFunctionsIcon.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Space-and-Functions&project=Space.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Multivariable-Space-and-Functions) <br> |
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| [**Calculus: Integrals**](https://www.mathworks.com/matlabcentral/fileexchange/105740-calculus-integrals) <br> | <img src="Images/IntegralsIcon.png" width="171" alt="IntegralsIcon.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/105740-calculus-integrals) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Calculus-Integrals&project=Integrals.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Calculus-Integrals) <br> |
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## Getting Started
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### Accessing the Module
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### **On MATLAB Online:**
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Use the [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj) link to download the module. You will be prompted to log in or create a MathWorks account. The project will be loaded, and you will see an app with several navigation options to get you started.
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Download or clone this repository. Open MATLAB, navigate to the folder containing these scripts and double\-click on [Integrals.prj](<matlab: openProject("Integrals.prj")>). It will add the appropriate files to your MATLAB path and open an app that asks you where you would like to start.
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Ensure you have all the required products ([listed below](#H_E850B4FF)) installed. If you need to include a product, add it using the Add\-On Explorer. To install an add\-on, go to the **Home** tab and select <img src="Images/AddOnsIcon.png" width="16" alt="AddOnsIcon.png"> **Add-Ons** > **Get Add-Ons**.
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# Scripts
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## [**LineIntegrals.m**](Scripts/LineIntegrals.m)
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## [MultipleIntegrals](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj&file=Scripts/MultipleIntegrals.mlx)
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| :-- | :-- | :-- |
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| <img src="Images/LineIntegral.png" width="171" alt="LineIntegral.png"> <br> | **In this script, students will...** <br> $\bullet$ Compute line integrals given a path and vector field <br> $\bullet$ Identify conservative vector fields and use this to compute line integrals <br> $\bullet$ Apply Green's Theorem to compute area <br> | **Academic disciplines** <br> $\bullet$ Electrical Engineering <br> $\bullet$ Physics <br> $\bullet$ Mathematics <br> |
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| <img src="Images/SlicePumpkin.png" width="171" alt="SlicePumpkin.png"> <br> | **In this script, students will...** <br> $\bullet$ Compute and visualize double and triple integrals <br> $\bullet$ Use change of variables to simplify and evaluate multiple integrals <br> | **Academic disciplines** <br> $\bullet$ Mathematics <br> |
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## **MultipleIntegrals.m** (planned)
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## [MultipleIntegralsApplications](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj&file=Scripts/MultipleIntegralsApplications.mlx)
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| :-- | :-- | :-- |
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| <img src="Images/IrregularMeasurement.png" width="171" alt="IrregularMeasurement.png"> <br> | **In this script, students will...** <br> $\bullet$ Compute and visualize double and triple integrals <br> $\bullet$ Use change of variables to simplify and evaluate multiple integrals <br> | **Academic disciplines** <br> $\bullet$ Physics <br> $\bullet$ Mathematics <br> |
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| <img src="Images/TiltedPrism.png" width="171" alt="TiltedPrism.png"> <br> | **In this script, students will...** <br> $\bullet$ Identify appropriate coordinate systems to describe scenarios <br> $\bullet$ Apply double and triple integrals to solving problems <br> | **Academic disciplines** <br> $\bullet$ Physics <br> $\bullet$ Wireless communications <br> $\bullet$ Mechanical Engineering <br> $\bullet$ Electrical Engineering <br> |
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## **SurfaceIntegrals.m** (planned)
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## [**LineIntegrals**](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Integrals&project=Integrals.prj&file=Scripts/LineIntegrals.mlx)
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| :-- | :-- | :-- |
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| <img src="Images/LineIntegral.png" width="171" alt="LineIntegral.png"> <br> | **In this script, students will...** <br> $\bullet$ Compute line integrals given a path and vector field <br> $\bullet$ Identify conservative vector fields and use this to compute line integrals <br> $\bullet$ Apply Green's Theorem to compute area <br> | **Academic disciplines** <br> $\bullet$ Electrical Engineering <br> $\bullet$ Physics <br> $\bullet$ Mathematics <br> |
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## **SurfaceIntegrals** (planned)
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| <img src="Images/ControlSurface.png" width="171" alt="ControlSurface.png"> <br> | **In this script, students will...** <br> $\bullet$ Explore the concepts of control surfaces and control volumes <br> $\bullet$ Use the divergence theorem <br> | **Academic disciplines** <br> $\bullet$ Electrical Engineering <br> $\bullet$ Physics <br> $\bullet$ Mathematics <br> |
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The license for this module is available in the [LICENSE.md](https://github.com/MathWorks-Teaching-Resources/Multivariable-Integrals/blob/release/LICENSE.md).
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# Related Courseware Modules
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| **Courseware Module** <br> | **Sample Content** <br> | **Available on:** <br> |
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| :-- | :-- | :-- |
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| [**Applied Partial Differential Equations**](https://www.mathworks.com/matlabcentral/fileexchange/172650-applied-partial-differential-equations) <br> | <img src="Images/image_15.png" width="171" alt="image_15.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/172650-applied-partial-differential-equations) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Applied-PDEs&project=AppliedPDEs.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Applied-PDEs) <br> |
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| [**Multivariable: Space and Functions**](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> | <img src="Images/image_18.png" width="159" alt="image_18.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Space-and-Functions&project=Space.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Multivariable-Space-and-Functions) <br> |
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| [**Applied Partial Differential Equations**](https://www.mathworks.com/matlabcentral/fileexchange/172650-applied-partial-differential-equations) <br> | <img src="Images/PDEsIcon.png" width="171" alt="PDEsIcon.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/172650-applied-partial-differential-equations) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Applied-PDEs&project=AppliedPDEs.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Applied-PDEs) <br> |
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| [**Multivariable: Space and Functions**](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> | <img src="Images/MultivariableFunctionsIcon.png" width="159" alt="MultivariableFunctionsIcon.png"> <br> | [<img src="Images/OpenInFX.png" width="91" alt="OpenInFX.png">](https://www.mathworks.com/matlabcentral/fileexchange/180356-multivariable-space-and-functions) <br> [<img src="Images/OpenInMO.png" width="136" alt="OpenInMO.png">](https://matlab.mathworks.com/open/github/v1?repo=MathWorks-Teaching-Resources/Multivariable-Space-and-Functions&project=Space.prj&file=README.mlx) <br> [GitHub](https://github.com/MathWorks-Teaching-Resources/Multivariable-Space-and-Functions) <br> |
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Or feel free to explore our other [modular courseware content](https://www.mathworks.com/matlabcentral/profile/authors/37969341).
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# Educator Resources
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- [Educator Page](https://www.mathworks.com/academia/educators.html)
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- [Educator Page](https://www.mathworks.com/academia/educators.html)
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# Contribute
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Looking for more? Find an issue? Have a suggestion? Please contact the [MathWorks Educator Content Development Team](mailto:%20onlineteaching@mathworks.com). If you want to contribute directly to this project, you can find information about how to do so in the [CONTRIBUTING.md](https://github.com/MathWorks-Teaching-Resources/Multivariable-Integrals/blob/release/CONTRIBUTING.md) page on GitHub.
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*©* Copyright 2025 The MathWorks, Inc
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