Area Between Curves

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

• Section 6.1 – Area Between Curves

Expected Educational Results

• 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.

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.

Application: Area Between Curves

Definition: Probability Density Function

$f\left(x\right)$$f(x)$ is a probability density function if all of the following conditions are met:

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

Definition: Probability

Suppose $x$$x$ is a continuous random variable where $f\left(x\right)$$f(x)$ is a probability density function, the the probability that $x$$x$ has values between $x=a$$x=a$ and $x=b$$x=b$ [denoted $P\left(a\le x\le b\right)$$P(a\leq x\leq b)$] is ${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\int_{a}^{b}{f(x)\,dx}$.

Definition: Normal Probability Density Function

Suppose $x$$x$ has mean $m$$m$ and standard deviation $s\ge 0$$s\geq 0$, then $f\left(x\right)=\frac{1}{s\sqrt{2\pi }}{e}^{-{\left(\frac{x-m}{s\sqrt{2}}\right)}^{2}}$$f\left(x\right)=\frac{1}{s\sqrt{2\pi}}e^{-\left(\frac{x-m}{s\sqrt{2}}\right)^{2}}$ is the normal probability density function (a.k.a., normal curve).

Investigation 10

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

Investigation 11

Let $m=10$$m=10$ and $s=1$$s=1$. Estimate (to two (2) decimal places) the following probabilities:

1. $P\left(9\le x\le 11\right)$$P(9\leq x\leq 11)$
2. $P\left(10\le x\le 13\right)$$P(10\leq x\leq 13)$
3. $P\left(7\le x\le 8\right)$$P(7\leq x\leq 8)$
4. $P\left(x\le 12\right)$$P(x\leq 12)$
5. $P\left(x\ge 11\right)$$P(x\geq 11)$

Use Technology to Compute Normal Probabilities

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:

x1(* Investigation 11-4 *)2Integrate[1/(1*Sqrt[2*Pi])/E^((x-10)/(1*Sqrt[2]))^2, {x,-Infinity,12}]//N

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Integrate[ ] has two arguments:

1. The function in the integrand of the average value integral;

2. $\left\{x,a,b\right\}$$\{x,a,b\}$, where

1. $x$$x$ is the independent variable
2. $a$$a$ is the lower limit of integration, i.e., the left-hand endpoint of the interval;
3. $b$$b$ is the upper limit of integration, i.e., the right-hand endpoint of the interval.
3. For help on using the Integrate[ ] function:

1. In Mathematica, execute the code: $\text{?Interate}$$\text{?Interate}$
2. Click on $\vee$$\or$ near the bottom-left of output
3. Click on local
4. Read how to use the Integrate[ ] function – you will be able to copy-paste code.
4. $\text{//N}$$\text{//N}$ at the end of the code forces Mathematica to return a decimal approximation.

5. You may need parens, $\left(\phantom{\rule{0.167em}{0ex}}$$(\,$ and $\phantom{\rule{0.167em}{0ex}}\right)$$\,)$, to group multiple terms in the numerator and denominator.

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

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