{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# 129 integ below is a partial table of integr als of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ulus. Gottfried Wilhelm Leibniz (1646-1716) in his manuscript dated 29 October 1673, was the first to use the integral sign ∫, the elongated or stretched S continuous as we know and use it today, to denote continuous summation. Leibniz called contin the expression ∫ f (x) dx the INTEGRAL from the Latin integralis o r whole. i ntegralis However, both the concepts of defining the r elationship between Differential and Integral Calculus: DERIVATIVE & ANTIDERIVATIVE and TANGENT and “ ar ea under the cur v e ”, are useful but limited. The correct concept that corresponds to the relationship is INSTANTANEOUS RATE OF CHANGE and CHANGE. 126 Over view Ov er vie w differentia We may define a WELL-BEHAVED function F(x) and on differentiation let : dif entiation d F(x) f(x) = dx Hence f(x) is the DERIVATIVE of F(x). If we write f(x)dx = dF(x) then we call f(x) the DIFFERENTIAL COEFFICIENT. inverse opera differentia We may now define the inver se oper ation of dif f er entiation as finding the in dif entiation ANTIDERIVATIVE . So : F(x) = ANTIDERIVATIVE { f(x)} differentia Just like with differentiation we may construct a table of ANTIDERIVATIVES. The dif entiation calculation becomes almost mechanical. This is purely from the calculation point calculation of view. From the Analysis point of view we may infer that : if f(x) is the expression of the Analysis INSTANTANEOUS RATE OF CHANGE of F(x), then F(x) must be the expression of CHANGE. From the Geometric point of view we have an even more interesting relationship. It Geometric seems almost magical. area under f(x) CHANGE in F(x) = = F(b) − F(a) over interval [a, b] over interval [a, b] Integral Calculus developed out of the need for a general method to find areas, volumes and centres of gravity. Computing the area under f(x) involves a continuous summation . It is from this process of continuous summation that continuous we have the operation expressed as integration derived from the Latin integralis integration integralis and the Mathematical notation or symbol ∫ to def...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern