# 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 1–1: I can state the Fundamental Theorem of Calculus.

• Objective 1–2: I can explain the meaning of the Fundamental Theorem of CalculusPart I.

• Objective 1–3: I can explain the meaning of the Fundamental Theorem of CalculusPart II.

• Objective 1–4: I can determine the properties of an area function using FTC-I and Calculus I.

• Objective 1–5: I can evaluate indefinite integrals using the Fundamental Theorem of CalculusPart I.

• Objective 1–6: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 1–7: 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 1–8: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.

• Objective 1–9: 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.

## Evaluating Definite Integrals

Recall,

Fundamental Theorem of Calculus - Part II [FTC-II]: Let $f\left(t\right)$$f(t)$ be continuous on $\left[a,b\right]$$[a,b]$. Let $F\left(t\right)$$F(t)$ be any antiderivative of $f\left(t\right)$$f(t)$. Then ${\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt=F\left(b\right)-F\left(a\right)$$\displaystyle \int_a^b{f(t)\,dt}=F(b)-F(a)$.

https://PollEv.com/multiple_choice_polls/0NpJB99oKCqWZiOuGVt0v/respond

#### Investigation 17

What does it mean for $f\left(t\right)$$f(t)$ to be continuous on $\left[a,b\right]$$[a,b]$? Explain.

#### Investigation 18

NOTE: Exact values are always expected on all Assessments. This Investigation asks for decimal approximations only to verify answers.

Evaluate each of the following integrals (re-write the integrand, if needed):

1. ${\int }_{0}^{1}\left({x}^{2}+x+1{\right)}^{2}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_0^1{(x^2+x+1)^2\,dx}$

2. ${\int }_{0}^{\pi }\mathrm{sin}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{0}^{\pi}{\sin{(x)}\,dx}$

3. ${\int }_{\pi /6}^{\pi /4}\frac{1+{\mathrm{cos}}^{2}\left(x\right)}{{\mathrm{cos}}^{2}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{\pi/6}^{\pi/4}{\frac{1+\cos^2{(x)}}{\cos^2{(x)}}\,dx}$

4. ${\int }_{-1}^{3}{e}^{x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{-1}^{3}{e^{x}\,dx}$

5. ${\int }_{2}^{3}\frac{{x}^{4}-1}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_2^3{\frac{x^4-1}{x^2-1}\,dx}$

6. ${\int }_{0}^{1/2}\frac{{x}^{2}-1}{{x}^{4}-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_0^{1/2}{\frac{x^2-1}{x^4-1}\,dx}$

7. ${\int }_{1}^{2}\left({x}^{-2}+\sqrt[5]{{x}^{2}}\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_1^2{\left(x^{-2}+\sqrt[5]{x^2}\right)\,dx}$

8. ${\int }_{0}^{1}\left(x+3\right)\left({x}^{2}+1\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_0^1{\left(x+3\right)\left(x^2+1\right)\,dx}$

9. ${\int }_{1}^{4}\frac{{x}^{2}+3x+5}{{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_1^4{\frac{x^2+3x+5}{x^2}\,dx}$

10. ${\int }_{7\pi /6}^{7\pi /4}\frac{1}{x\phantom{\rule{0.167em}{0ex}}\sqrt{{x}^{2}-1}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{7\pi/6}^{7\pi/4}{\frac{1}{x\,\sqrt{x^2-1}}\,dx}$

11. ${\int }_{0}^{3\pi /4}\mathrm{sin}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_0^{3\pi/4}{\sin{(x)}\,dx}$

12. ${\int }_{-\pi /3}^{\pi /3}\mathrm{sin}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{-\pi/3}^{\pi/3}{\sin{(x)}\,dx}$

13. ${\int }_{-\pi /6}^{5\pi /3}\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{-\pi/6}^{5\pi/3}{\cos{(x)}\,dx}$

14. ${\int }_{\pi /3}^{\pi /4}\frac{\mathrm{sin}\left(x\right)}{{\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{\pi/3}^{\pi/4}{\frac{\sin{(x)}}{\cos^2{\!(x)}}\,dx}$

15. ${\int }_{0}^{\pi /6}\frac{\mathrm{sin}\left(2x\right)}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_{0}^{\pi/6}{\frac{\sin{(2x)}}{\cos{(x)}}\,dx}$

16. ${\int }_{0}^{\pi /4}\frac{\mathrm{sec}\left(x\right)}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int_0^{\pi/4}{\frac{\sec{(x)}}{\cos{(x)}}\,dx}$

#### Use Technology to Verify Definite Integrals

Mathematica

NOTE: Mathematica is used here to verify definite integrals. On all Assessments, you must show that you understand the Calculus concept of the definite integral.

​x1(* Investigation 22-03 *)2(* Find definite integral [1+cos^2(x)]/cos^2(x) *)3(* from x=pi/6 to x=pi/4 *)4
5Integrate[(1+Cos[x]^2)/Cos[x]^2, {x, Pi/6, Pi/4}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. Integrate[ ] has two arguments:

1. The integrand, i.e., function of the definite integral;

2. $\left\{x,a,b\right\}$$\{x,a,b\}$, where $x$$x$ is the variable used in the integrand; $a$$a$ is the lower limit of integration; $b$$b$ is the upper limit of integration.

3. The Mathematica syntax for ${e}^{x}$$e^x$​​ is $\text{E}\phantom{\rule{-0.167em}{0ex}}\wedge \phantom{\rule{-0.167em}{0ex}}x$$\text{E}\!\wedge\!x$​​; ${\mathrm{cos}}^{2}\left(x\right)$$\cos^2{(x)}$​​ is $\text{Cos[x]}\phantom{\rule{-0.167em}{0ex}}\wedge \phantom{\rule{-0.167em}{0ex}}2$$\text{Cos[x]}\!\wedge\!2$​​, etc.

4. Remember to use parens $\left($$\left(\right.$ and $\right)$$\left.\right)$​ to group multiple terms in the numerator or denominator of a rational expression.

5. Remember, correct Mathematica code will be all black except for variables.

6. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

NOTE: Python is used here to verify definite integrals. You must show that you understand the Calculus I concept of the definite integral.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.

2. The Python syntax for ${e}^{x}$$e^x$ is $\text{exp(x)}$$\text{exp(x)}$; ${\mathrm{cos}}^{2}\left(x\right)$$\cos^2{(x)}$ is $\text{cos(x)**2}$$\text{cos(x)**2}$; ${x}^{4}$$x^4$ is $\text{x**4}$$\text{x**4}$, etc.

3. Python requires $\ast$$*$ for explicit multiplication.

4. You only need to change the three lines of code that are between the two $\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}\mathrm{#}$$\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#$ lines.​

5. Remember to use parens $\left($$\left(\right.$ and $\right)$$\left.\right)$​ to group multiple terms in the numerator or denominator of a rational expression.

6. To execute code, press the “View the result” button:

#### Investigation 19

Explain the difference between an indefinite integral and a definite integral.

## Homework

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

• Section 5.3: # 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49;

• Section 5.4: # 27, 29, 31, 33, 35, 37, 39, 41, 45, 47, 49, 51, 53.