Integration in One Dimension

Students entering into upper-division physics courses are typically familiar and comfortable with integration as taught in mathematics courses. In physics, there is additional language and interpretation which accompany integration. By reintroducing integration early in upper-division courses, many common student difficulties which arise in electricity and magnetism and other physics courses can be addressed.

Students may be encountering new approaches to integration which are not addressed in math courses. For example, Internal Energy of the "Derivative Machine" may be the first time students have experimentally measured an integral and can be used to encourage fluency between multiple representations of integration such as numerical, graphical, and algebraic representations. Additionally, students typically integrate with respect to the variable of the function such as $x$ in $f(x)=\int_{x_0}^{x_f}(mx+b)dx$. However, oftentimes in physics the resulting function is of a different variable such as $g(m)=\int_{x_0}^{x_f}(mx+b)dx$ which students may not have encountered in previous calculus and introductory physics courses.

This sequence can serve as a quick reintroduction to integration and can likely be completed entirely in less than one hour.


  • Internal Energy of the "Derivative Machine" (Estimated time: ): This small group activity serves as an introduction to experimentally measuring an integral by determining the internal energy, $U$, of a nonlinear system at several locations. Students must choose a point of zero internal energy and then add up, incrementally, the force at each small change in distance by numerical integration.
  • 1D Rectangular Integration (Estimated time: ): This lecture introduces students to the language which physicists use with regard to integration. This includes the notion of integration as “chopping and adding” and “accumulation” as well as explanations of boundaries and functions as used in physics.
  • Functions (Estimated time: ): This lecture or homework problem addresses how the parameter which the integration is occurring over isn't necessarily the parameter which is varied following the integration. In mathematics, the resulting function from an integral has variables which are the same over which the integration occurred. For example, if integrating over $dx$, the function following the integration will be $f(x)$. However, in physics, often another parameter is varied following the integration, so if integrating over $dx$, the function may be $g(m)$. By addressing this through a homework problem or a lecture, students will have the knowledge to recognize when this occurs in more complex physical problems.

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