Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Fundamental Theorem of Calculus

Definition: Area Under a Curve f(x)

The area under a curve f(x) is the area between the curve and the x-axis.

Definition: Net Area Under a Curve f(x)

The net area under a curve f(x) is the net area between the curve and the x-axis on the interval [a,b], where

net area=area under f(x) above x-axisarea under f(x) below x-axis

and is represented as abf(x)dx.

Theorem: Fundamental Theorem of Calculus - Part I (FTC-I)

Fundamental Theorem of Calculus - Part I: Let f(t) be continuous on [a,b] and for any x in [a,b], let F(x)=axf(t)dt. Then F(x) is differentiable on [a,b] and F(x)=f(x), for any x in [a,b].

NOTE: The variable x is the independent variable, and the variable t is called a dummy variable.

Explanation of FTC-I

Conditions

There are two (2) condition(s), i.e., requirements, for this theorem:

Conclusion

If both conditions are true, then the two (2) statements in the conclusion (the statements after the word “then” in the theorem) are also true, i.e.,

Interpretation of FTC-I

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Created: Tuesday, 25 August 2020 02:28 EDT Last Modified: Monday, 15 August 2022 - 15:32 (EDT)