# 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 CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart 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 CalculusPart I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\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\left(x\right)$$f(x)$ be continuous on $\left[a,b\right]$$[a,b]$ and $F\left(x\right)$$F(x)$ any antiderivative of $f\left(x\right)$$f(x)$, then

${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=F\left(b\right)-F\left(a\right)$$\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\left(x\right)$$f(x)$ is continuous on the closed interval $\left[a,b\right]$$[a,b]$;

• $F\left(x\right)$$F(x)$ is any antiderivative of $f\left(x\right)$$f(x)$, i.e., ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$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.,

${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=F\left(b\right)-F\left(a\right)$$\displaystyle \int_a^b{f(x)\,dx}=F(b)-F(a)$.

#### Interpretation of FTC-II

• ${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}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\left(x\right)$$f(x)$ over the closed interval $\left[a,b\right]$$[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.