Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

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.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Evaluating Definite Integrals

Recall,

Fundamental Theorem of Calculus - Part II [FTC-II]: Let f(t) be continuous on [a,b]. Let F(t) be any antiderivative of f(t). Then abf(t)dt=F(b)F(a).

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Investigation 17

What does it mean for f(t) to be continuous on [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. 01(x2+x+1)2dx

  2. 0πsin(x)dx

  3. π/6π/41+cos2(x)cos2(x)dx

  4. 13exdx

  5. 23x41x21dx

  6. 01/2x21x41dx

  7. 12(x2+x25)dx

  8. 01(x+3)(x2+1)dx

  9. 14x2+3x+5x2dx

  10. 7π/67π/41xx21dx

  11. 03π/4sin(x)dx

  12. π/3π/3sin(x)dx

  13. π/65π/3cos(x)dx

  14. π/3π/4sin(x)cos2(x)dx

  15. 0π/6sin(2x)cos(x)dx

  16. 0π/4sec(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.

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

  3. The Mathematica syntax for ex​​ is Ex​​; cos2(x)​​ is Cos[x]2​​, etc.

  4. Remember to use parens ( and )​ 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 ex is exp(x); cos2(x) is cos(x)**2; x4 is x**4, etc.

  3. Python requires for explicit multiplication.

  4. You only need to change the three lines of code that are between the two ########################### lines.​

  5. Remember to use parens ( and )​ 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:

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