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A planimeter is a device used to measure the area enclosed by a curve and the most familiar is the polar planimeter. Proofs using Green's theorem for the rolling and polar planimeters are illustrated, which can be used in a vector calculus course. Both these planimeters are available in mechanical and electronic versions for commercial use.

This article introduces students to the study of fractional calculus and derivatives. Problems involving fractional derivatives of exponential functions, trigonometric functions, and iterated integrals are presented.

Gerardus Mercator's map-projection technique of the 16th century can be used to solve the calculus problem of multivariable arc length. Mercator's map enabled rhumb lines, which are constant bearing paths connecting two points. These rhumb lines can be parametrized using rectangular coordinates and trigonometric identities. An improper integral then gives arc length in real three-dimensional space.

Concavity and convexity of functions in calculus can be defined without resort to derivatives. This approach uses the concept of lower bounding line. Convexity is first defined and then the definition is expanded to include concavity and differentiability. Five conditions, one lemma and one theorem are used.

Summation by parts can be applied to calculus and number theory. This technique is the discrete analog of integration by parts and can be applied to geometric figures composed of areas of rectangles. The technique is applicable to many topics in calculus including the definite integral, geometric series, arithmetic sequences, derivatives of powers and reduction formulae. From number theory, sequences of Fibonacci numbers are examined.

Existence of divergent samples of 'harmonic-like' series is proved. An uncountably infinite family of examples of the divergence of balanced harmonic like series is constructed.

A solution to the problem in which a bug inside a vessel begins to crawl upward when a liquid is poured onto the vessel is provided. It is shown that the speed is the product of a geometric quantity and the flow rate of the liquid likewise speed can also be expressed as an explicit function of time.

Constrained-optimization and related-rates word problems are biggest problems and methods suggested as solution are contrasted in first-year calculus. It is concluded that a different approach to constrained-optimization problems, similar to that widely used for related-rates problem would be advantageous.

It is reported that the In calculus and the In function is introduced by an integral and its derivative is then found by the means of the Fundamental Theorem of Calculus. The method of placing the natural logarithm and the exponential function on an equal footing is explained.

A theorem presents conditions for the intervals supporting a prescribed area arise in statistics to be the shortest. Further, the fundamental theorem of calculus shows that the rate of change of area at each x is f(x).

The language of predicate calculus is used to examine different possible meanings that Lincoln might have invented to convey. A question is raised on the interpretation that Lincoln had in his mind.

The numerous tests done on a dog-named Elvis, to study whether dogs make use of calculus, are presented. Various diagrams and tables showing the results are mentioned.

An evidence in support of the vibrant vitality of calculus is presented by providing a result that links together two theorems, one intuitive and other nonintuitive. The two theorems linked are the Fubini's theorem on exchanging the order of integration and the equality of mixed partial derivatives.

The concept of convergent series being good and divergent series bad is altered. An example is presented to help undergraduate students in calculus, numerical analysis, and real analysis, as well as professionals unfamiliar with the subject.

An expression is derived for the total volume of rain absorbed in terms of the speed of the traveler. It will be helpful for a calculus class to show that maximums and minimums are not always at the points where the slope of the function is zero.