Student Showcase: Real-World Applications of Double and Triple Integrals

As part of Concordia’s advanced Multivariable Calculus course, our high school students recently engaged in a powerful learning experience focused on the applications of double and triple integrals, a unit that blended conceptual depth, inquiry-based exploration, and real-world relevance.

Each topic within the unit began with a problem set rooted in inquiry, encouraging students to ask questions, make predictions, and uncover the mathematical structures behind complex ideas. From exploring volumes over irregular domains to modeling surface area and hypervolumes, students developed a deep understanding of integration in multiple dimensions.

Throughout the unit, students tackled core concepts such as:

• Double integrals over rectangular and irregular regions
• Coordinate system transformations (polar, cylindrical, spherical)
• Surface area and volume modeling in 3D space
• Triple integrals and their applications in scientific contexts

Some students took advantage of virtual reality headsets to enhance their understanding of cylindrical and spherical coordinate systems, using immersive visualizations to reinforce spatial intuition and conceptual clarity.

Throughout the unit, students revisited key calculus foundations, including limits, volume integration over regular and irregular domains, surface area, and coordinate transformations. They then applied these tools to self-designed investigations, resulting in innovative and interdisciplinary projects such as:

• A predictive model for baseball performance using volume integrals to estimate hit probabilities.
• An environmental study calculating surface-level CO₂ concentrations across atmospheric layers.
• A machine learning project applying gradient descent and image analysis to forecast balding patterns.
• A cartographic model projecting spherical surfaces onto 2D maps using integral-based transformations.

The final poster presentations allowed students to synthesize their research, demonstrate mathematical modeling, and communicate their findings to peers, teachers, and the wider community.

"This unit exemplifies deep, applied learning in STEM pursuing original ideas, making higher mathematics come alive in meaningful ways," says Dr. Fujiwara, STEM Coordinator at Concordia Shanghai.