# 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.

## Area Function

### Definition: Area Function

According to the definition of a Definite Integral [Section 5.2 of the textbook and discussed in Math 2211 – Calculus of a Single Variable - I], ${\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\int_a^b{f(t)\,dt}$ is the Net Area under the curve $f\left(t\right)$$f(t)$, where

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

The following DESMOS activity will help you graphically understand the Fundamental Theorem of Calculus - Part I:

Recall,

Fundamental Theorem of Calculus - Part I [FTC-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]$.

### Check Conditions of FTC-I

• $f\left(t\right)$$f(t)$ is the black graph. $f\left(t\right)$$f(t)$ is continuous on $\left[0,12\right]$$[0,12]$. NOTE: A function is continuous on an interval IF:

• there are NO holes in the graph;

• there are NO jump discontinuities in the graph; and

• there are NO infinite discontinuities in the graph.

#### Procedure

Let $g\left(x\right)={\int }_{0}^{a}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$g(x)=\int_0^a{f(t)\,dt}$.

NOTE: I used $g\left(x\right)$$g(x)$ rather than $F\left(x\right)$$F(x)$, the letter used to represent functions is arbitrary.

In the embedded DESMOS page above,

1. Move the slider for the parameter $a$$a$ [Box 3] to values $a=0,0.5,1,1.5,\dots ,11,11.5,12$$a = 0, 0.5, 1, 1.5, \ldots, 11, 11.5, 12$.

2. Identify the value of ${\int }_{0}^{a}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\int_0^a{f(t)\,dt}$ [Box 2] and compare to the area [shaded blue and red and the value listed in the Table [Box 5] under $f\left(t\right)$$f(t)$ [black curve].

3. What happens to $g\left(x\right)$$g(x)$ from $a=7$$a=7$ to $a=7.5$$a=7.5$? Explain.

#### Activity 05

1. What does the $y$$y$-value, i.e., for each green point refer to? Explain.

2. Explain what happens to the green curve, i.e., $g\left(x\right)$$g(x)$, when $a$$a$ increases.

#### Activity 06

Using Calculus I, determine which of the following is true:

• The black curve, $f\left(x\right)$$f(x)$, is the derivative of the green curve, $g\left(x\right)$$g(x)$. Explain.

• The green curve, $g\left(x\right)$$g(x)$, is the derivative of the black curve, $f\left(x\right)$$f(x)$. Explain.

### Homework

At this time, you should be able to complete the following assignments:

• Section 5.3: # 5.