Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyFundamental Theorem of CalculusDefinition: Area Under a Curve

**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.

The **area under a curve**

The **net area under a curve**

and is represented as

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

**NOTE**: The variable **independent** variable, and the variable *dummy* variable.

**Conditions**

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

is continuous for all values on$f(x)$ ;$[a,b]$ is defined as a$F(x)$ **net area function**, i.e.,$F(x)={\int}_{a}^{x}f(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$

**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.,

is differentiable on$F(x)$ ;$[a,b]$ , for any${F}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)=f(x)$ in$x$ .$[a,b]$

is a function of$F(x)$ ;$x$ is a${\int}_{a}^{x}f(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ **sum**of a large number of very small values; andthe lower limit of integration is a constant,

;$a$ the upper limit of integration is the variable

$x$

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

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