Fundamental Theorem of Calculus (FTC)

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

Expected Educational Results

• 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}^{\prime }(x)\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\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.

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.

Fundamental Theorem of Calculus

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

Fundamental Theorem of Calculus - Part II: Let $f(x)$$f(x)$ be continuous on $[a,b]$$[a,b]$ and $F(x)$$F(x)$ any antiderivative of $f(x)$$f(x)$, then

${\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a)}$$\displaystyle \int_a^b{f(x)\,dx}=F(b)-F(a)$.

Explanation of FTC-II

Conditions

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

• $f(x)$$f(x)$ is continuous on the closed interval $[a,b]$$[a,b]$;

• $F(x)$$F(x)$ is any antiderivative of $f(x)$$f(x)$, i.e., $F^{\prime }(x)=f(x)$$F^{\,\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.,

${\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a)}$$\displaystyle \int_a^b{f(x)\,dx}=F(b)-F(a)$.

Interpretation of FTC-II

• ${\displaystyle {\int }_a^{b}f(x)dx}$$\displaystyle \int_a^b{f(x)\,dx}$ is called a 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 $f(x)$$f(x)$ over the closed interval $[a,b]$$[a,b]$, including:

  • 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.

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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)