*Two of Euler's familiar identities, the sum of reciprocals of squares of integers and the infinite product expression of sine function, are proved again.*

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*Every standard calculus textbook contains the derivations for the definite integral of \(x\) and \(x^2\) using Riemann sums \(\ldots\)*

*The classical Buffon needle problem is to find the probability that a needle of length \(n\) when dropped on a floor made of boards of width \(b\) will cross a crack between the boards.*

*Geometrical notions are abundant in calculus, where one learns how problems involving them can be addressed via the derivative or integral.*

*There are few random processes more avidly watched than the state lottery drawings in which six numbered balls are chosen from a set of forty-four.*

*"Imagine a fuel tank in the shape of a horizontal cylinder, with length \(l\) and circular cross section with diameter \(d \ldots\)"*

*Most calculus students take it for granted that the shell and disk methods for computing the volume of a solid of revolution must always give the same result.*

*In this capsule proofs of the equivalence of the two definitions of the Fibonacci numbers are discussed. This helps the undergraduate view mathematics as a unified whole with a variety of...*

*A study of the sensitivity of the volume of cans to the radius of the base*

*"This problem also fits quite nicely in the area of convex geometry, from which we draw the following proof..."*