# 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

### Definition: Area Under a Curve $f\left(x\right)$$f(x)$

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

### Definition: Net Area Under a Curve $f\left(x\right)$$f(x)$

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

$\displaystyle \text{net area}={\color{green}\text{area under } f(x) \text{ above }x\text{-axis}}-{\color{red}\text{area under } f(x) \text{ below }x\text{-axis}}$

and is represented as ${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$\displaystyle\int_a^b{f(x)\,\text{d}x}$.

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

Fundamental Theorem of Calculus - Part I: Let $f\left(t\right)$$f(t)$ be continuous on $\left[a,b\right]$$[a,b]$ and for any $x$$x$ in $\left[a,b\right]$$[a,b]$, let $F\left(x\right)={\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle F(x)=\int_a^x{f(t)\,dt}$. Then $F\left(x\right)$$F(x)$ is differentiable on $\left[a,b\right]$$[a,b]$ and ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x)=f(x)$, for any $x$$x$ in $\left[a,b\right]$$[a,b]$.

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

#### Explanation of FTC-I

Conditions

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

• $f\left(x\right)$$f(x)$ is continuous for all values on $\left[a,b\right]$$[a,b]$;

• $F\left(x\right)$$F(x)$ is defined as a net area function, i.e., $F\left(x\right)={\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle F(x)=\int_a^x{f(t)\,dt}$. More on this later.

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.,

• $F\left(x\right)$$\displaystyle F(x)$ is differentiable on $\left[a,b\right]$$[a,b]$;

• ${F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)=f\left(x\right)$$F^{\,\prime}(x)=f(x)$, for any $x$$x$ in $\left[a,b\right]$$[a,b]$.

#### Interpretation of FTC-I

• $F\left(x\right)$$\displaystyle F(x)$ is a function of $x$$x$;

• ${\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_a^x{f(t)\,dt}$ is a sum of a large number of very small values; and

• the lower limit of integration is a constant, $a$$a$;

• the upper limit of integration is the variable $x$$x$

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