Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyEvaluating Definite IntegralsInvestigation 17Investigation 18Use Technology to Verify Definite IntegralsInvestigation 19HomeworkCC BY-NC-SA 4.0

**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

**Objective 1–1**: I can state the*Fundamental Theorem of Calculus*.**Objective 1–2**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part I*.**Objective 1–3**: I can explain the meaning of the*Fundamental Theorem of Calculus*–*Part 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 Calculus*–*Part I*.**Objective 1–6**: I can evaluate definite integrals using the*Fundamental Theorem of Calculus*–*Part II*.**Objective 1–7**: I can use the*Net Change Theorem*to identify the meaning of .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **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.

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.

Recall,

**Fundamental Theorem of Calculus - Part II [FTC-II]**: Let

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

What does it mean for

**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):

${\int}_{0}^{1}({x}^{2}+x+1{)}^{2}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{\pi}\mathrm{sin}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{\pi /6}^{\pi /4}\frac{1+{\mathrm{cos}}^{2}(x)}{{\mathrm{cos}}^{2}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{-1}^{3}{e}^{x}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{2}^{3}\frac{{x}^{4}-1}{{x}^{2}-1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{1/2}\frac{{x}^{2}-1}{{x}^{4}-1}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{1}^{2}({x}^{-2}+\sqrt[5]{{x}^{2}}){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{1}(x+3)({x}^{2}+1){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{1}^{4}\frac{{x}^{2}+3x+5}{{x}^{2}}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{7\pi /6}^{7\pi /4}\frac{1}{x{\textstyle \phantom{\rule{0.167em}{0ex}}}\sqrt{{x}^{2}-1}}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{3\pi /4}\mathrm{sin}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{-\pi /3}^{\pi /3}\mathrm{sin}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{-\pi /6}^{5\pi /3}\mathrm{cos}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{\pi /3}^{\pi /4}\frac{\mathrm{sin}(x)}{{\mathrm{cos}}^{2}{\textstyle \phantom{\rule{-0.167em}{0ex}}}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{\pi /6}\frac{\mathrm{sin}(2x)}{\mathrm{cos}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ ${\int}_{0}^{\pi /4}\frac{\mathrm{sec}(x)}{\mathrm{cos}(x)}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$

**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```
```

5`Integrate[(1+Cos[x]^2)/Cos[x]^2, {x, Pi/6, Pi/4}]`

**Warnings**:

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.*Integrate[ ]*has two arguments:The integrand, i.e., function of the definite integral;

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

The

*Mathematica*syntax for is${e}^{x}$ ;$\text{E}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\wedge {\textstyle \phantom{\rule{-0.167em}{0ex}}}x$ is${\mathrm{cos}}^{2}(x)$ , etc.$\text{Cos[x]}{\textstyle \phantom{\rule{-0.167em}{0ex}}}\wedge {\textstyle \phantom{\rule{-0.167em}{0ex}}}2$ Remember to use parens

and$($ to group multiple terms in the numerator or denominator of a rational expression.$)$ Remember, correct

*Mathematica*code will be all black except for variables.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**:

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.The

*Python*syntax for is${e}^{x}$ ;$\text{exp(x)}$ is${\mathrm{cos}}^{2}(x)$ ;$\text{cos(x)**2}$ is${x}^{4}$ , etc.$\text{x**4}$ *Python*requires for explicit multiplication.$\ast $ You only need to change the three lines of code that are between the two

lines.$\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}\mathrm{\#}$ Remember to use parens

and$($ to group multiple terms in the numerator or denominator of a rational expression.$)$ To execute code, press the “View the result” button:

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

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.

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**Created**: Tuesday, 25 August 2020 02:28 EDT
**Last Modified**: Friday, 19 August 2022 - 19:08 (EDT)