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Objectives

  • Describe the meaning of the Mean Value Theorem for Integrals.
  • State the meaning of the Fundamental Theorem of Calculus, Part 1.
  • Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
  • State the meaning of the Fundamental Theorem of Calculus, Part 2.
  • Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
  • Explain the relationship between differentiation and integration.

Summary

We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus says that if f is a continuous function on [ a , b ] and F is an antiderivative of f , then

∫ a b f ( x ) d x = F ( b ) − F ( a ) .

Hence, if we can find an antiderivative for the integrand f , evaluating the definite integral comes from simply computing the change in F on [ a , b ] .

See the Desmos Demonstration on Evaluating Definite Integrals.

A slightly different perspective on the FTC allows us to restate it as the Total Change Theorem, which says that

∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) ,

for any continuously differentiable function f . This means that the definite integral of the instantaneous rate of change of a function f on an interval [ a , b ] is equal to the total change in the function f on [ a , b ] .

See the Desmos Demonstration.

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