During Class

Brain Gains

Definite Integrals and Geometric arguments

Use a geometric argument to find the value of each integral. If needed, start by drawing the region over the specified bounds. Then use Mathematica to check your solution is correct.

1. $\displaystyle\int_2^5 7 dx$

2. $\displaystyle\int_0^4 f(x) dx$ where $f(x) = \begin{cases}3x & 0\leq x\leq 1 \\ 4-x & 1\leq x \leq 4\end{cases}$

3. $\displaystyle\int_0^4 \sqrt{16-x^2} dx$

4. $\displaystyle\int_0^{10} \sqrt{25-(x-5)^2} dx$

Group Meeting

Teach One Another

Give each group member 1-2 minutes to share what they prepared from the between class work.

Work on the chalkboard as you complete these activities and exercises, and pass the chalk as you finish each one. Leave up your work so that you can compare your answers with your neighbors. If you notice an answer differs from a neighboring group's, then have a discussion with them.

Antiderivatives and Indefinite Integrals

1. Find a function $H_1(x)$ whose derivative is $h(x) = 4x^3$. Then find a second function $H_2(x)$ whose derivative is $h(x)=4x^3$. We call $H_1$ and $H_2$ antiderivatives of $h(x)$, because they undo the differentiation process.
2. Find two different functions whose derivative is $e^{-x}$.

If $F$ is an antiderivative of $f$, then both $G(x) = F(x) + 11$ and $H(x) = F(x) - 3$ are also antiderivatives of $f$. Notice $F(x) \neq G(x)$ and $F(x) \neq H(x)$ but the derivatives of these functions are equal. In fact, if $C$ is any real number and $F$ is an antiderivative of $f$, then $F(x)+C$ is an antiderivative of $f$.

3. Compute $\int 3x dx$. Use the Mathematica code below to check your answer.

Integrate[3x,x] in Mathematica to check your answer. 

4. Compute $\int x^2-4x dx$. Use Mathematica to check your answer.
5. Compute $\int 5+\frac{1}{x^2} dx$. Use Mathematica to check your answer.

Activity - Accumulation Function

1. Consider the function $\ds g(x) = \begin{cases}\frac{3}{8}x^2 - \frac{3}{2}x + \frac{3}{2} & 0 < x < 2 \\ 0 & \text{otherwise}\end{cases}.$ Let $g(x)$ be a function that defines the top of a target. A dart is randomly thrown at the target, and we let $X$ be the random variable that records the $x$-coordinate of the dart's location.

Use Mathematica to perform any definite integrals.

1. What is the total area of the target, so the area under $g$ and above the $x$-axis?
2. Find the pdf $f(x)$ of $X$.
3. Find the cdf $F(x)$ of $X$.
4. Graph $F(x)$.

Integrate[(3/8)*x^2 - (3/2)*x + (3/2), {x, 0, 2}]

Integrate[(3/8)*x^2 - (3/2)*x + (3/2), {x, 0, x}]

F <- function(x){
  ifelse(x<0, 0, 
         ifelse(x>2,1, 
                1/8*x^3 - 3/4*x^2 + 3/2*x))
}


x <- seq(-2,10,0.1)
plot(x,F(x),type='l')

F = Piecewise[{
   {0, x <= 0},
   {1/8*x^3 - 3/4*x^2 + 3/2*x, x < 2},
   {1, x >= 2}
   }]
Plot[F, {x, -2, 10}]

2. Consider the same scenario as before, but this time let $\ds g(x) = \begin{cases}e^{-\lambda x} & x \geq 0 \\ 0 & \text{otherwise}\end{cases}$ for some $\lambda >0$.

Use Mathematica to perform any definite integrals.

1. What is the total area of the target, so the area under $g$ and above the $x$-axis?
2. Find the pdf $f(x)$ of $X$.
3. Find the cdf $F(x)$ of $X$.
4. Graph $F(x)$.

Discussion

Antiderivatives, Indefinite Integrals, Accumulation Functions

Consider the pdf $f(x) = e^{-x} $ for $x \geq 0$ and zero otherwise.

• We can verify that $F_1(x) = -e^{-x}$, $F_2(x) = 5 - e^{-x} $, and $F_3(x) = -e^{-x} - 2$ are all antiderivatives of $f$. Are any of these function the CDF of the random variable $X$?
• Find the cdf of $X$.
  • We know $F(x) = P(X \leq x) = \int_{-\infty}^x f(t)dt$.

$$F(x) = \int_{-\infty}^x f(t)dt = \begin{cases} \int_{-\infty}^x 0dt = 0 & x < 0 \\ \\ \int_{0}^x e^{-t}dt = 1-e^{-x} & x \geq 0 \end{cases}.$$

$Assumptions = x > 0;
Integrate[Exp[-t],{t,0,x}]

Integrate[Exp[-x],x]

• We know that $F(x)$ is from the family of functions $\int e^{-x}dx$. How do we pick which one?
  • We need to identify the constant. We know $F(0) = 0$.

• The cdf $F(x) = \int_{-\infty}^x f(t)dt$ is an accumulation function (calculating the accumulation of probability).
  • More generally an accumulation function is of the form $F(x) = \int_{a}^x f(t)dt$ where $a$ is a constant.
  • An accumulation function is one antiderivative of $f$. The one where $F(a) = 0$.

Summary

• A definite integral is a number.
  • Example: $\int_{0.5}^2 e^{-x} dx \approx 0.4712$. (exact value of this number is $e^{-0.5} - e^{-2}$)
• An indefinite integral is a family of functions (collection of antiderivatives).
  • Example: $\int e^{-x} dx = -e^{-x} + C$.
• An accumulation function is an antiderivative.
  • Example: $\int_0^x e^{-t} dt = 1-e^{-x}$. (The antiderivative when $F(0) = 0$. Just one function from the family of antiderivatives.)
  • What do you think happens when we take the derivative of an accumulation function?

Activity - Making Connections using CDFs and PDFs

1. Consider a random variable $X$ with probability density function $f(x) = \frac{1}{5}$ for $2\leq x\leq 7$.

1. Use a geometric argument to compute $P(3\leq X\leq 5)$. Then use Mathematica to verify your answer is correct by writing down an appropriate definite integral and computing it.
2. Use a geometric argument to state the CDF $F(x)$. Then use Mathematica to verify your answer is correct by writing down an appropriate definite integral and computing it.
3. Now that we have a formula for $F(x)$, use that formula to compute $F(3)$ and $F(5)$. Then use $F(3)$ and $F(5)$ to obtain $P(3\leq X\leq 5)$.

2. Consider a random variable $X$ with probability density function $f(x) = \frac{2x}{25}$ for $0\leq x\leq 5$.

1. Use a geometric argument to compute $P(1\leq X\leq 3)$. Then use Mathematica to verify your answer is correct by writing down an appropriate definite integral and computing it.
2. Use a geometric argument to state the CDF $F(x)$. Then use Mathematica to verify your answer is correct by writing down an appropriate definite integral and computing it.
3. Now that we have a formula for $F(x)$, use that formula to compute $F(1)$ and $F(3)$. Then use $F(1)$ and $F(3)$ to obtain $P(1\leq X\leq 3)$.

3. Consider a random variable $X$ with probability density function $g(x) = \frac{2x}{9}$ for $0\leq x\leq 3$.

1. State a function $G(x)$ whose derivative is $g(x)$. Then state another. Then state another. How many such functions are there?
2. Compute $G(x)-G(0)$.
3. Use Mathematica to compute the cumulative distribution function $\int_0^x g(x)dx$.
4. Compute $P(1\leq X\leq 2)$ using the function $G(x)$.

4. Consider a random variable $X$ with probability density function $g(x) = \frac{3x^2}{64}$ for $0\leq x\leq 4$.

1. State a function $G(x)$ whose derivative is $g(x)$. Then state another. Then state another. How many such functions are there?
2. Compute $G(x)-G(0)$.
3. Use Mathematica to compute the cumulative distribution function $\int_0^x g(x)dx$.
4. Compute $P(1\leq X\leq 3)$ using the function $G(x)$.

5. Consider a random variable $X$ with probability density function $g(x) = \frac{2x}{21}$ for $2\leq x\leq 5$.

1. State a function $G(x)$ whose derivative is $g(x)$. Then state another. Then state another. How many such functions are there?
2. Compute $G(x)-G(2)$.
3. Use Mathematica to compute the cumulative distribution function $\int_2^x g(x)dx$.
4. Compute $P(3\leq X\leq 4)$ using the function $G(x)$.

6. Consider a random variable $X$ with probability density function $g(x) = \frac{x^2}{39}$ for $2\leq x\leq 5$.

1. State a function $G(x)$ whose derivative is $g(x)$. Then state another. Then state another. How many such functions are there?
2. Compute $G(x)-G(2)$.
3. Use Mathematica to compute the cumulative distribution function $\int_2^x g(x)dx$.
4. Compute $P(3\leq X\leq 4)$ using the function $G(x)$.