A restatement of the Fundamental Theorem of Calculus is presented in this lesson along with a corollary that is used to find the value of a definite integral analytically.

Restating the Fundamental Theorem

You have discovered the Fundamental Theorem in the context of finding areas under a curve but a more general version of this theorem can be proved without an appeal to area. The following is a restatement of the Fundamental Theorem.

If f is continuous on [a, b], then the function

has a derivative at every point in [a, b], and the derivative is

That is, the derivative of a definite integral of f whose upper limit is the variable x and whose lower limit is the constant a equals the function f evaluated at x. This is true regardless of the value of the lower limit a. The function named F is the same as the area function that was previously explored.

Using the Restated Fundamental Theorem

• Set Angle mode to Radian
• Execute NewProb from the Clean Up menu

16.3.1 Use the restatement of the Fundament theorem to evaluate the following derivatives, then check your predictions with the TI-89.

Click here for the answer.

16.3.2 Predict the following derivative. Check your answer with the TI-89. Hint: You will have to use the chain rule.

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16.3.3 Evaluate the following derivative and check with your TI-89.

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16.3.4 Find a more general version of the Fundamental theorem by predicting the following derivative. Check your work with the TI-89.

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Using a Corollary of the Fundamental Theorem

The following corollary of the Fundamental theorem gives a method for evaluating a definite integral.

Corollary

If f is continuous on [a,b], then

, where .

The function F is called an antiderivative of the function f.

16.3.5 Use the corollary to predict the value of , then check your work with the TI-89.

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16.3.6 Use the corollary of the Fundamental theorem to evaluate then check your work with your calculator.

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