Definite Integrals and Area

A collection of mathematical and physical problems ranging from early childhood addition to advanced network theory.

A comprehensive unit for 12th Grade Calculus students focusing on the integration of polar functions to find area, arc length, and surface area. Students transition from Cartesian thinking to radial accumulation, mastering the geometry of circular sectors and polar coordinate transformations.

An advanced 11th-grade Calculus unit focusing on the integration of parametric and polar coordinate systems. Students analyze motion, calculate complex areas, perform error analysis, and complete a final synthesis project based on particle kinematics.

This calculus sequence guides 11th-grade students through the integration techniques required to calculate area and arc length within polar coordinate systems. From the geometric derivation of the polar sector formula to complex multi-curve regions and boundary measurements, students apply integral calculus to circular geometries.

This sequence explores calculus in the polar coordinate system, focusing on differentiation and integration. Students will master finding slopes of tangent lines, calculating areas of polar regions and intersection areas, and determining arc lengths of polar curves.

This sequence explores the intersection of calculus and geometry through infinite series and fractals. Students investigate convergence and divergence using visual area models, fractal dimensions, and physical simulations like block stacking.

A graduate-level exploration of the Calculus of Variations, focusing on optimizing functionals. Students derive the Euler-Lagrange equation and apply it to physics and geometry problems like the Brachistochrone and Isoperimetric challenges.

A comprehensive sequence for undergraduate calculus students exploring the construction, convergence, and real-world utility of power series. Students move from the technical mechanics of convergence tests to applying Taylor series in physics and engineering contexts.

An advanced graduate sequence exploring vector calculus from 3D fields to differential forms on manifolds, focusing on fluid dynamics and electromagnetic theory. It moves from parameterizing static fields to understanding global topological constraints on curved surfaces.

A project-based calculus sequence for 12th grade students focusing on the engineering applications of vector-valued functions, including path optimization, differentiability, and arc length.

A comprehensive unit connecting differentiation and integration through the Fundamental Theorem of Calculus. Students transition from visualizing accumulation to mastery of algebraic evaluation, applying these concepts to real-world net change and total area problems.

This sequence guides students through the conceptual transition from geometry to calculus by investigating the Area Problem. Students begin by estimating areas under curves using geometric shapes, discovering the relationship between rectangle width and approximation accuracy. As the sequence progresses, learners formalize these approximations using Sigma notation and limits, ultimately defining the definite integral.

Une leçon complète sur le calcul intégral appliqué à l'économie, couvrant l'intégration par parties, l'indice de Gini et les surplus du consommateur et du producteur.

A focused study on applying the Fundamental Theorem of Calculus to evaluate functions defined by integrals. Students will practice finding specific values by integrating and evaluating at given boundaries.

Students investigate how regular polygons with increasing numbers of sides eventually 'converge' into a circle, using the apothem formula to derive the area of a circle.

A 12th-grade calculus prep lesson that bridges algebraic symmetry with integral properties, teaching students to use even and odd function characteristics to simplify definite integrals.

A specialized AP Calculus lesson exploring the unique geometric and analytical properties of Euler's number. Students use graphing software to discover why e is the unique base where the function's height, slope, and area under the curve are identical.

A high-school level lesson for AP Calculus and Statistics students focusing on using Desmos for complex integrals and statistical calculations, emphasizing the balance between manual understanding and technological efficiency.