Area Between CurvesExpected Educational ResultsBloom’s TaxonomyApplication: Area Between CurvesDefinition: Probability Density FunctionDefinition: ProbabilityDefinition: Normal Probability Density FunctionInvestigation 10Investigation 11Check Your WorkUse Technology to Compute Normal ProbabilitiesCC BY-NC-SA 4.0

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

**Section 6.1**– Area Between Curves

**Objective 10–1**: I can set up the integral to compute the area of the region bounded by two or more curves.**Objective 10–2**: I can evaluate the integral to compute the area of the region bounded by two or more curves using FTC-II.**Objective 10–3**: I can use technology to estimate the area of the region bounded by two or more curves using FTC-II.

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.

**probability density function** if all of the following conditions are met:

$0\le f(x)$ - at least one tail (one “side” of the curve) never touches the
-axis$x$ ${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx=1$

Suppose **probability** that

Suppose

- Use the above embedded DESMOS graph to verify the normal curve is a probability density function.

Let

$P(9\le x\le 11)$ $P(10\le x\le 13)$ $P(7\le x\le 8)$ $P(x\le 12)$ $P(x\ge 11)$

**NOTE**: On any assessments, you may use technology to estimate normal probabilities without showing any work.

**Mathematica**

To check the answer in **Investigation 11-4**:

x

1`(* Investigation 11-4 *)`

2`Integrate[1/(1*Sqrt[2*Pi])/E^((x-10)/(1*Sqrt[2]))^2, {x,-Infinity,12}]//N`

**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: http://www.jjw3.com/TECH_Common_Functions.pdf.*Integrate[ ]*has two arguments:The function in the integrand of the average value integral;

, where$\{x,a,b\}$ is the independent variable$x$ is the lower limit of integration, i.e., the left-hand endpoint of the interval;$a$ is the upper limit of integration, i.e., the right-hand endpoint of the interval.$b$

For help on using the

*Integrate[ ]*function:- In
*Mathematica*, execute the code:$\text{?Interate}$ - Click on
near the bottom-left of output$\vee $ - Click on local
- Read how to use the
*Integrate[ ]*function – you will be able to copy-paste code.

- In
at the end of the code forces$\text{//N}$ *Mathematica*to return a decimal approximation.You may need parens,

and$({\textstyle \phantom{\rule{0.167em}{0ex}}}$ , to group multiple terms in the numerator and denominator.$\phantom{\rule{0.167em}{0ex}}})$ 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.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Created**: Thursday, 17 September 2020 2:48 EDT
**Last Modified**: Wednesday, 02 March 2022 - 03:58 (EST)