Fundamental Theorem of Calculus: Makes a connection between Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”) Historically, indefinite integration has always been defined to be the inverse of differentiation. is the collection of all possible anti-derivatives of f(x), which happen to differ only by a constant. But definite integration, motivated by the problem of finding areas under curves, was originally defined as a limit of Riemann sums. Is the limit of any Riemann sum as the number of rectangles approaches infinity … provided the limit of the Lower and the limit of the Upper Riemann sums are equal. Sec4-4: (Day1) Fundamental Theorem of Calculus Sec4-4: #2-38 evens

Only later was it discovered that the limits of these Riemann sums can actually be computed with antiderivatives, leading to our modern Fundamental Theorem of Calculus. _______________________________ _________________ The fundamental theorem allows us to calculate definite integrals By using anti-derivatives (indefinite integrals) _________________

Examples: Applying the Fundamental Theorem of Calculus Section 4-4

Examples: Applying the Fundamental Theorem of Calculus