The modern practice in school curriculum is that instructors teach differentiation first before integration in contrast to the inception of calculus (Pitt, 2012).
In the 17th century, Newton and Leibniz contribution made a milestone in advance of scientific developments in Europe. Kepler found the volume of a bucket of wine to be a pile of thin horizontal circular discs. B. Cavalieri’s line formed from many points, plane formed from many lines and solid made from many planes. All these gave initiating ideas in integration. At the same time Newton and Leibniz related integration and differentiation to be reverse processes in their “Fundamental Theorem of Calculus” popularly known as “Newton-Leibniz Formula” (Stewart, 2010).
The calculus method became a very powerful tool in solving mathematical problems where other analytical methods failed those times. For instance, Jacob Bernoulli using calculus methods discovered that logarithmic spring under various transformations is invariant. In 1696, Johann Bernoulli asked a question that said, assuming friction was zero the path taken by a point mass to slide from a point that is higher to a point that is lower taking the shortest time possible (Hawkins, 2001). This led to the development of the calculus of variation. Other 17th century developments are contained in Euler’s introduction in analysis infinitorum, institutiones calculi differentialis and institutions calculi Integralis.
Consider the following shape, to find the volume and the surface area of the shape using integration then employ the use of infinitesimally small areas and volumes. Obtain the volume using this formula
For example, take a parabola in the x-y plane whose height bases on another parabolic function. Say, from x=-2 to x=2 that has elevation determined by a circle of radius 4 i.e. . Taking y to lie between y=x2+4 and y=12-x2 then the volume is given by
Integrating again using ...
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