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5.c – The Fundamental Theorem of Calculus and Definite Integrals

5.c – The Fundamental Theorem of Calculus and Definite Integrals. Examples. The definite integral of f ( x ) from x = a to x = b is denoted. f ( x ) is called the integrand , a the lower limit , and b the upper limit . . The First Fundamental Theorem of Calculus.

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5.c – The Fundamental Theorem of Calculus and Definite Integrals

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  1. 5.c – The Fundamental Theorem of Calculus and Definite Integrals

  2. Examples The definite integral of f (x)from x = a to x = b is denoted f(x) is called the integrand, a the lower limit, and b the upper limit.

  3. The First Fundamental Theorem of Calculus If f is continuous on [a, b], then the function g is defined by where F is the general antiderivative of f, that is, a function such that F ' = f.

  4. Basic Properties of the Indefinite Integral Let a, b, and c be constants and f and g be continuous functions on [a, b].

  5. Examples Evaluate:

  6. Definite Integrals With The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Properties of Odd and Even: Suppose f is continuous on [– a, a].

  7. Examples - Evaluate Evaluate by changing your limits of integration to values that are in terms of u.

  8. First Fundamental Theorem of Calculus (Alternate Definition) We’ve shown that represents the antiderivative of f with respect to x. It follows that the derivative of the antiderivative should return the original function (that is, the integrand). Upper Limit Must Be Variable Part Lower Limit Must Be Numerical Part t is called a dummy variable as it just holds a place.

  9. Examples Use WolframAlpha confirm the first part of the First Fundamental Theorem of Calculus with the following examples.

  10. Examples Evaluate the following within one minute.

  11. First Fundamental Theorem of Calculus – Part 1 (Generalized) If f is continuous on [a, b] and u is an unknown, differentiable function of x, then

  12. Examples Use WolframAlpha to determine the following:

  13. Examples Evaluate

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