Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyFundamental Theorem of CalculusTheorem: Fundamental Theorem of Calculus - Part II (FTC-II)Explanation of FTC-IIInterpretation of FTC-IICC BY-NC-SA 4.0

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

**Section 4.9**– Antiderivatives**Section 5.3**– The Fundamental Theorem of Calculus**Section 5.4**– Indefinite Integrals and the Net Change Theorem

**Objective 01–01**: I can state the*Fundamental Theorem of Calculus*.**Objective 01–02**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–03**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–04**: I can determine the properties of an area function using*FTC-I*and*Calculus I*.**Objective 01–05**: I can evaluate indefinite integrals using the*Fundamental Theorem of Calculus*–*Part I*.**Objective 01–06**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 01–07**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **Objective 01–08**: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.**Objective 01–09**: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

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.

**Fundamental Theorem of Calculus - Part II**: Let

**Conditions**

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

is continuous on the closed interval$f(x)$ ;$[a,b]$ is any antiderivative of$F(x)$ , i.e.,$f(x)$ .${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$

**Conclusion**

If **both** conditions are **true**, then the conclusion (the statement after the word “then” in the theorem) is also true, i.e.,

is called a${\int}_{a}^{b}f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ **definite integral**;A definite integral is a

**sum**of a large number of very small values;The result of a definite integral is the

**net change**of over the closed interval$f(x)$ , including:$[a,b]$ net area “under” a function;

area of a region bounded between two curves;

volume of a solid formed by rotating a bounded region about an axis;

length of a curve;

area bounded by a polar curve;

work performed on object;

etc.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Created**: Tuesday, 25 August 2020 02:28 EDT
**Last Modified**: Monday, 30 May 2022 - 13:58 (EDT)