Browse Classroom Capsules and Notes

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Algebraic Set Operations, Multifunctions, and Indefinite Integrals

Careless use of the set of antiderivatives, coupled with integration by parts, leads to such absurdities as 0 = 1.

Proof without Words: Integration by Parts

The rule of integration by parts is proven visually.

Proof without Words: The Reflection Property of the Parabola

The reflection property of the parabola is demonstrated visually.

A Coordinate Approach to the AM-GM Inequality

A short proof is given for AM-GM inequality by a convex exponential function.

A Fundamental Theorem of Calculus for All Riemann Integrable Functions

The author weakens the conditions usually seen in a calculus text for the fundamental theorem of calculus.

Proof without Words: Alternating Sum of Squares = Triangular Number

A visual proof of the evaluation of the alternating sum of squares is given.

Building the Cheapest Sandbox: An Allegory with Spinoff

With a real-world optimization problem, the author shows a fun application of calculus concerning extreme values, highlighting differences between the academia viewpoint and the practical...

Euler's Other Proof

This is a short proof of a famous result of Euler about summation of the following series: \(\sum 1/{n^2} = {\pi^2}/6\).

Sledge-Hammer Integration

The author finds an area-preserving sequence of functions, which is increasingly flattened and tends to a constant determined by the integral of the given function.

Trick or Technique?

The author studies the functions whose integral can be evaluated by an algebra trick using one-step or two-step integration by parts.