Development of calculus ideas began in ancient Greek times starting Archimedes in around 225 BC used the exhaustion method, that was previously developed by Zeno, Eudoxus and others, to prove that the area of a segment of parabola can be derived from knowing the area of circumscribed parallelogram. This finding relates to integral calculus.
In the early 17th century, Descartes created a new method of finding the slope of a normal line. Next, Barrow was working on how to find the slope of a tangent line to the circle by shortening the two sets of points on a chord in a circle. This was known as Barrow's differential triangle. Barrow also may have understood about the inverse relationship between differentiation and calculus.
The development of integral and derivative calculus was fairly separate until the 17th century that Barrow, Isaac Newton and Gottfried Leibniz discovered the relationship between the two branches of the field and were able to write a proof for this theorem. Knowing this relationship is extremely helpful in finding the integral formulas that we use to find curved areas.
There were many other people not listed here or as well known as Archimedes, Leibniz, and Newton, but they all created a progression of events to create this enormously useful and elegant mathematical tool.
Applications of calculus have expanded from finding areas of curved shapes to biology to profit optimization and on to engineering applications.
Topics include the discovery of infinity concepts by Zeno, area approximation methods, Descartes' approach to finding normal lines, and finally Newton and Leibniz proving The Fundamental Theorem of Calculus. Time period discussed is from the 5th century www-gap.dcs.st-and.ac.uk/../../The_rise_of_calculus.html
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