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# (PART) Integral Calculus {-}
# Antiderivatives
## Lecture Content
[**Video: The Indefinite Integral**](https://youtu.be/noxadE3ZoJY)
[**Video: Integration by Substitution**](https://youtu.be/mUOC5tahsaA)
[**Video: Integration by Parts**](https://youtu.be/Ovl4B0bYA5A)
---
## Lecture Notes
### Basic Antiderivatives and Rules
Antiderivatives are the reverse of derivatives. The antiderivative of a function \( f(x) \) is a function \( F(x) \) such that:
\[
F'(x) = f(x)
\]
The indefinite integral of \( f(x) \) is:
\[
\int f(x) \, dx = F(x) + C
\]
Where \( C \) is the constant of integration.
#### Rules of Antidifferentiation
- **Sum Rule:**
\[
\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx
\]
- **Constant Multiple Rule:**
\[
\int c \cdot f(x)\,dx = c \cdot \int f(x)\,dx
\]
- **Power Rule:**
\[
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \text{ for } n \ne -1
\]
- **Exponential Rule:**
\[
\int e^x \,dx = e^x + C
\]
- **Reciprocal Rule:**
\[
\int \frac{1}{x} \,dx = \ln|x| + C
\]
#### Common Antiderivatives
| Function \( f(x) \) | Antiderivative \( \int f(x)\,dx \) |
|------------------------------|-------------------------------------------|
| \( x^n \) (for \( n \ne -1 \)) | \( \frac{x^{n+1}}{n+1} + C \) |
| \( \frac{1}{x} \) | \( \ln|x| + C \) |
| \( e^x \) | \( e^x + C \) |
| \( a^x \) | \( \frac{a^x}{\ln a} + C \) |
| \( \sin x \) | \( -\cos x + C \) |
| \( \cos x \) | \( \sin x + C \) |
| \( \sec^2 x \) | \( \tan x + C \) |
| \( \sec x \tan x \) | \( \sec x + C \) |
---
### Integration by Substitution
Substitution is used when the integrand contains a composite function.
#### Steps:
1. **Choose a substitution:**
Let \( u = g(x) \), where \( g(x) \) is an inner function.
2. **Differentiate \( u \):**
Compute \( du = g'(x)\,dx \).
3. **Rewrite the integral:**
Replace all \( x \)-terms with \( u \)-terms.
4. **Integrate in terms of \( u \).**
5. **Substitute back:**
Replace \( u \) with the original expression.
**Example:**
\[
\int 2x \cos(x^2) \, dx
\]
Let \( u = x^2 \), then \( du = 2x \, dx \).
So the integral becomes:
\[
\int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C
\]
---
### Integration by Parts
Use this method when the integrand is a product of two functions.
#### Formula:
\[
\int u\,dv = uv - \int v\,du
\]
#### Steps:
1. **Choose \( u \)** and **\( dv \)** from the integrand.
- Use **LIPET** (Logarithmic, Inverse trig, Polynomial, Exponential, Trig) as a guide for choosing \( u \).
2. **Differentiate \( u \)** to get \( du \), and **integrate \( dv \)** to get \( v \).
3. **Apply the formula:**
\[
\int u\,dv = uv - \int v\,du
\]
4. **Simplify and integrate again if necessary.**
**Example:**
\[
\int x e^x \, dx
\]
Let \( u = x \), \( dv = e^x dx \)
Then \( du = dx \), \( v = e^x \)
\[
\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C
\]
---
## Examples
### Antiderivatives
[**Video: Solutions**](https://www.youtube.com/watch?v=Sv1gz4CeAv4)
1. Consider the function $f(x) = 6x^{10} + 2x^6 - 4x^4 - 4$. Find an antiderivative of \( f(x) \).
2. Evaluate the definite integral $\int_5^8 \frac{1}{x^2} \, dx$.
3. Find the indefinite integral $\int \left(\frac{4}{x^5} + 3x + 5 \right) \, dx$.
4. Compute the indefinite integral $\int (x - 4)(x + 4) \, dx$.
5. Find the indefinite integral $\int \left(-5x^4 + \frac{3}{x} - \frac{3}{x^2} - 4\sqrt{x} \right) \, dx$.
6. Consider the function $f(t) = 10 \sec^2(t) - 3 t^2$. Let \( F(t) \) be the antiderivative of \( f(t) \) such that \( F(0) = 0 \). Find $F(3)$.
7. Find a function \( f(x) \) such that $f'(x) = 4 e^x + 3x$ and $f(0) = -5$.
---
### U-Substitution
[**Video: Solutions**](https://www.youtube.com/watch?v=3UfJA1iZ0xk)
1. Evaluate the integral $\int x^6 (x^7 - 11)^{50} \, dx$.
2. Evaluate the indefinite integral $\int x^3 \sqrt{11 + x^4} \, dx$.
3. Evaluate the indefinite integral $\int \frac{x^5}{x^6 + 2} \, dx$.
4. Evaluate the indefinite integral $\int \frac{6}{x \ln(9x)} \, dx$.
5. Evaluate the indefinite integral $\int 3 e^{3x} \sin\bigl(e^{3x}\bigr) \, dx$.
6. Evaluate the indefinite integral $\int \frac{x + 2}{x^2 + 4x + 5} \, dx$.
7. Evaluate the indefinite integral $\int x^6 e^{x^7} \, dx$.
---
### Integration by Parts
[**Video: Solutions**](https://www.youtube.com/watch?v=GT6KpNf4sfU)
1. Use integration by parts to evaluate the definite integral $\int_1^e 3t^2 \ln t \, dt$.
2. Evaluate the indefinite integral $\int 6x e^x \, dx$.
3. Evaluate the indefinite integral $\int 6x e^{4x} \, dx$.
4. Use integration by parts to evaluate the integral $\int 8z \cos(4z) \, dz$.
5. Use integration by parts to evaluate the integral $\int x^6 e^x \, dx$.
---
## Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
1. Evaluate the following integrals:
a. $$\int (x + x^3) \, dx$$
b. $$\int (\sin u + \cos u) \, du$$
c. $$\int \sqrt[4]{x} + \sin x \, dx$$
d. $$\int \frac{1}{x} + \frac{2}{x^2} \, dx$$
2. Evaluate the following integrals using the substitution method:
a. $$\int (2x+5)^{-3} \, dx$$
b. $$\int \sqrt{4x-5} \, dx$$
c. $$\int \sin(4u+6) \, du$$
d. $$\int x e^{-x^2+1} \, dx$$
3. Evaluate the following integrals using integration by parts:
a. $$\int 2x e^{x} \, dx$$
b. $$\int (x^2 - 1) 3^{-x} \, dx$$
c. $$\int x^{2} \ln x \, dx$$
d. $$\int x^{3} (x+1)^{10} \, dx$$
4. (Applied) The marginal cost of producing the $x$th box of CDs is given by
$$10 - \frac{x}{(x^{2} + 1)^{2}}.$$
The total cost to produce 2 boxes is \$1,000. Find the total cost function $C(x)$.
5. (Applied) The number of housing starts in the United States can be approximated by
$$n(t) = \frac{1}{12} \left(1.1 + 1.2 e^{-0.08 t} \right) \text{ million homes per month } (t \geq 0)$$
where $t$ is time in months from the start of 2006. Find an expression for the total number $N(t)$ of housing starts in the US from January 2006 to time $t$.
---
## Self Assessment
Time yourself and try to solve the following questions within twenty minutes.
1. Evaluate the following integrals:
a. $$\int \left(\frac{1}{v^2} + \frac{2}{v}\right) dv$$
b. $$\int \frac{|u|}{u} + \sec^{2} u \, du$$
2. Evaluate the following integrals using the substitution method:
$$\int \left( 2x e^{x^{2} + 4} + \frac{5}{2x + 8} \right) dx$$
3. Evaluate the following integrals using integration by parts:
$$\int x \ln(2x) \, dx$$
4. The marginal cost of producing the $x$th box of Zip disks is
$$10 + \frac{x^{2}}{100,000}$$
and the fixed cost is \$100,000. Find the cost function $C(x)$.
---
## Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.
\bigskip
\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.}
\end{align*}
| Skill | D | CON | COM |
|------------------------------------------|---|-----|-----|
| Evaluate an integral using basic techniques. | | | |
| Evaluate an integral using the substitution method. | | | |
| Evaluate an integral using integration by parts. | | | |
| Solve applied problems using antiderivatives. | | | |
---
# Riemann Sums
## Lecture Content
[**Video: Riemann Sums and the Definite Integral**](https://youtu.be/zPqfFEdDXFQ)
---
## Lecture Notes
### Estimating Area Using \( n \) Rectangles
To estimate the area under a curve \( f(x) \) on an interval \([a, b]\), divide the interval into \( n \) subintervals of equal width:
\[
\Delta x = \frac{b - a}{n}
\]
Choose either **left endpoints** or **right endpoints** to evaluate the function:
- **Left Endpoint Approximation (LRAM):**
\[
A \approx \sum_{i=1}^{n} f(x_{i-1}) \Delta x
\]
- **Right Endpoint Approximation (RRAM):**
\[
A \approx \sum_{i=1}^{n} f(x_i) \Delta x
\]
Where:
\[
x_i = a + i\Delta x, \quad x_{i-1} = a + (i-1)\Delta x
\]
These methods give a numerical estimate of the area under the curve.
---
### Properties of Summation Notation
For any constants \( c \), and functions \( f(i) \), \( g(i) \):
1. **Linearity:**
\[
\sum_{i=1}^{n} [f(i) + g(i)] = \sum_{i=1}^{n} f(i) + \sum_{i=1}^{n} g(i)
\]
\[
\sum_{i=1}^{n} c \cdot f(i) = c \cdot \sum_{i=1}^{n} f(i)
\]
2. **Additive Indexing:**
\[
\sum_{i=m}^{n} f(i) = f(m) + f(m+1) + \dots + f(n)
\]
---
### Summation Formulas
These formulas are useful for simplifying Riemann sum expressions:
\[
\sum_{i=1}^{n} 1 = n
\]
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]
\[
\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
\]
\[
\sum_{i=1}^{n} i^3 = \left( \frac{n(n+1)}{2} \right)^2
\]
---
### Limit Definition of the Definite Integral
The definite integral of \( f(x) \) from \( a \) to \( b \) is defined as the limit of Riemann sums:
\[
\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x
\]
Where:
- \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval,
- \( x_i^* \) is any sample point in the \( i \)th subinterval \([x_{i-1}, x_i]\).
This definition captures the **exact area under the curve** as the number of rectangles \( n \) increases without bound and their width approaches zero.
---
## Examples
[**Video: Solutions**](https://youtu.be/6WUpRNbCM3w?si=Jz2sSDJfwhFOJU64&t=281)
1. Approximate the area under the curve \( y = x^3 \) from \( x = 2 \) to \( x = 5 \) using a Right Endpoint approximation with 6 subdivisions.
2. Estimate the area under the graph of $f(x) = x^2 + 6x + 12$ over the interval \([0, 2]\) using five approximating rectangles and right endpoints.
3. Evaluate the Riemann sum for $f(x) = \ln(x) - 0.7$ over the interval \([1, 3]\) using four subintervals, taking the sample points to be right endpoints.
---
## Practice Problems
Practice the techniques discussed in class and in the online videos by solving the following examples.
1. Use a Riemann sum to estimate the area under $f(x) = x^2$ on [1,5] using $n = 4$.
2. Use a Riemann sum to estimate the area under $f(x) = 2x+3$ on [-2,3] using $n=5$.
3. Use a Riemann sum to estimate the area under $f(x) = \frac{1}{1+x}$ on [0,1] using $n=4$.
4. Express in terms of $n$:
$$\sum_{i=1}^{n}(i + i^{2})$$
5. Express in terms of $n$:
$$\sum_{i=1}^{n}(3i - 5 + i^{3})$$
6. Express in terms of $n$:
$$\frac{1}{n} \sum_{i=1}^{n} \left( \frac{2i}{3n} + \frac{3i^{3}}{2n^{3}} - \frac{4i^{2}}{7n^{2}} \right)$$
7. Use the limit definition of the integral to evaluate
$$\int_{1}^{3} (2x+3) \, dx$$
8. Use the limit definition of the integral to evaluate
$$\int_{0}^{3} (x + x^2) \, dx$$
9. (Applied) A race car has a velocity of
$$v(t) = 600(1 - e^{-0.5t})$$
ft/s, $t$ seconds after starting. Use a Riemann sum with $n = 10$ to estimate how far the car has traveled in the first 4 seconds.
10. (Applied) The velocity of a stone moving under gravity $t$ seconds after being thrown up at 4 m/s is given by
$$v(t) = -9.8t + 4$$
m/s. Use a Riemann sum with 5 subdivisions to estimate
$$\int_0^1 v(t) \, dt.$$
---
## Self Assessment
Time yourself and try to solve the following questions within twenty minutes.
1. Use a Riemann sum to estimate the area under $f(x) = 4-x^{2}$ on $[0,2]$ using $n=4$.
2. Express in terms of $n$:
$$\frac{2}{n}\sum_{i=1}^{\infty}(2-3i^{2})$$
3. Evaluate:
$$\lim_{n \to \infty} \sum_{i=1}^{\infty} \left( \frac{i}{2n^{2}} - \frac{i^{2}}{3n^{3}} + \frac{4i^{3}}{n^{4}} \right)$$
4. Use the limit definition of the integral to evaluate
$$\int_0^2 (4-x^{2}) \, dx$$
5. The rate of U.S. per capita sales of bottled water for the period 2000–2008 can be approximated by
$$s(t) = 0.04 t^2 + 1.5 t + 17 \quad (0 \leq t \leq 8)$$
where $t$ is the time in years since the start of 2000. Use a Riemann sum with $n = 5$ to estimate the total U.S. per capita sales of bottled water from the start of 2003 to the start of 2008.
---
## Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.
\bigskip
\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.}
\end{align*}
| Skill | D | CON | COM |
|-----------------------------------------|---|-----|-----|
| Estimate areas using approximating rectangles. | | | |
| Manipulate expressions involving summation notation. | | | |
| Use the limit definition of the integral. | | | |
| Solve applied problems using Riemann sums. | | | |
---
# The Fundamental Theorem of Calculus
## Lecture Content
[**Video: The Fundamental Theorem of Calculus**](https://youtu.be/BVOSn1Gwqhs)
---
## Lecture Notes
### Fundamental Theorem of Calculus – Part I
Let \( f \) be a continuous function on the interval \([a, b]\), and define a function:
\[
F(x) = \int_a^x f(t)\,dt
\]
Then \( F \) is continuous on \([a, b] \), differentiable on \( (a, b) \), and:
\[
F'(x) = f(x)
\]
**Interpretation:**
The derivative of the area function \( F(x) \) gives back the original function \( f(x) \). This connects **differentiation** and **integration**.
---
### Fundamental Theorem of Calculus – Part II
Let \( f \) be continuous on \([a, b]\), and let \( F \) be any antiderivative of \( f \) on that interval. Then:
\[
\int_a^b f(x)\,dx = F(b) - F(a)
\]
**Interpretation:**
To evaluate a definite integral, find **any** antiderivative \( F \), and compute the difference \( F(b) - F(a) \). This allows us to compute exact areas without using limits or Riemann sums.
---
### Properties of Definite Integrals
Let \( f(x) \), \( g(x) \) be continuous on \([a, b]\), and let \( c \) be a constant. The following properties hold:
1. **Linearity:**
\[
\int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx
\]
\[
\int_a^b c \cdot f(x)\,dx = c \cdot \int_a^b f(x)\,dx
\]
2. **Additivity over Intervals:**
\[
\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx
\]
3. **Reversing Limits:**
\[
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx
\]
4. **Zero-Width Interval:**
\[
\int_a^a f(x)\,dx = 0
\]
5. **Comparison Property (if \( f(x) \ge g(x) \)):**
\[
\int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx
\]
6. **Absolute Value Inequality:**
\[
\left| \int_a^b f(x)\,dx \right| \le \int_a^b |f(x)|\,dx
\]
---
### Fundamental Theorem of Calculus
[**Video: Solutions**](https://www.youtube.com/watch?v=euIHL2RSWOI)
1. Evaluate the definite integral by interpreting it in terms of areas $\int_2^7 (2x - 12) \, dx$.
2. Use the limit definition of the integral to evaluate $\int_2^4 (x^2 + 4x - 3) \, dx$.
3. Evaluate the integral $\int_2^3 (x + 5)(x - 3) \, dx$
4. If $f(x) = \int_x^{19} t^6 \, dt,$ then find $f'(x)$
5. If $f(x) = \int_4^x t^3 \, dt,$ then find $f'(x)$. Also evaluate $f'(5)$.
6. If $f(x) = \int_{-2}^x^3 t^2 \, dt,$ then find $f'(x)$.
7. Use Part I of the Fundamental Theorem of Calculus to find the derivative of $f(x) = \int_4^x \left( \frac{1}{4} t^2 - 1 \right)^8 \, dt$. Find $f'(x)$.
---
## Practice Problems
1. Evaluate the following integrals:
a. $\displaystyle \int_{-3}^{3} \left(x + x^3\right) dx$
b. $\displaystyle \int_{0}^{\pi} \left(\sin u + \cos u\right) du$
c. $\displaystyle \int_{-1}^{1} \left(3x^4 - 2x^{-2} + x^{-5} + 4\right) dx$
d. $\displaystyle \int_{2}^{3} \left(2xe^{x^2 + 4} + \dfrac{5}{2x+8} \right) dx$
e. $\displaystyle \int_{0}^{1} \frac{e^x + e^{-x}}{2} dx$
2. Find the derivative of:
a. $g(x) = \displaystyle \int_{0}^{x} \left(u +u^3\right) du$
b. $g(x) = \displaystyle \int_{-5}^{x^2} \sqrt{1 + u^3} du$
c. $g(x) = \displaystyle \int_{\ln(x)}^{e^x} \left(u^6 - 1/u^2\right) du$
3. **(Applied)** The marginal revenue of the xth box of flash cards sold is $100e^{-0.001x}$. Find the revenue from items 101 through 1,000.
4. **(Challenge)** Differentiate
$$\int_{g(x)}^{h(x)} f(t) dt.$$
5. **(Challenge)** Show that
$$2 \leq \int_{0}^2 \sqrt{1 + x^3} dx \leq 6.$$
---
## Self Assessment
Time yourself and try to solve the following within 20 minutes:
1. $\displaystyle \int_{0}^{\pi/2} \left(1 + x + \cos x\right) dx$
2. $\displaystyle \int_{-1}^{1} x(x^2+1)^{10} dx$
3. $g(x) = \displaystyle \int_{x^3}^{\pi} e^{u^3 + 4u} du$
4. $g(x) = \displaystyle \int_{x}^{\pi/2} (1 + u + \cos u) du$
5. Given $C(x) = 246.76 + \int_{0}^x 5t dt$, find the fixed cost and marginal cost at $x=10$.
---
## Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.
\bigskip
\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.}
\end{align*}
| Skill | D | CON | COM |
|-----------------------------------------|---|-----|-----|
| Apply the FTCI to integrals. | | | |
| Apply the FTCII to integrals. | | | |
| Solve applied problems using the FTC. | | | |
---
# (PART) Additional (Important) Topics {-}
# Area between Curves
## Lecture Content
[**Video: Area Between Curves**](https://youtu.be/PfYcOdCgygk)
---
## Lecture Notes
### Definite Integrals as Net Area
The definite integral \( \int_a^b f(x) \, dx \) represents the **net area** between the graph of \( f(x) \) and the \( x \)-axis over the interval \([a, b]\):
- If \( f(x) \ge 0 \) on \([a, b]\), the integral gives the total area above the \( x \)-axis.
- If \( f(x) \le 0 \) on \([a, b]\), the integral gives the negative of the area below the \( x \)-axis.
- If \( f(x) \) changes sign on \([a, b] \), the integral gives the **net area**: area above minus area below.
**Note:** To find the **total area**, regardless of sign, use:
\[
\text{Total area} = \int_a^b |f(x)| \, dx
\]
---
### Simple Improper Integrals
An **improper integral** arises when:
1. The interval is **infinite**, e.g., \( \int_1^\infty \frac{1}{x^2} \, dx \)
2. The integrand is **unbounded** at a point in the interval, e.g., \( \int_0^1 \frac{1}{\sqrt{x}} \, dx \)
#### Infinite Interval Example:
\[
\int_1^\infty \frac{1}{x^2} \, dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left( -\frac{1}{x} \Big|_1^b \right) = 1
\]
#### Unbounded Function Example:
\[
\int_0^1 \frac{1}{\sqrt{x}} \, dx = \lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}} \, dx = \lim_{a \to 0^+} (2\sqrt{x} \big|_a^1) = 2
\]
An improper integral **converges** if the limit exists, and **diverges** if the limit does not exist.
---
### Area Between Two Curves
To find the area between two curves \( f(x) \) and \( g(x) \) on \([a, b]\), where \( f(x) \ge g(x) \):
\[
\text{Area} = \int_a^b [f(x) - g(x)] \, dx
\]
**Steps:**
1. **Find points of intersection:** Solve \( f(x) = g(x) \) to find the limits of integration.
2. **Identify top and bottom functions** on the interval.
3. **Set up and evaluate the integral:**
- If necessary, split the interval where the curves switch order.
4. For vertical slices (functions of \( y \)), use:
\[
\int_{c}^{d} [f(y) - g(y)] \, dy
\]
**Example:**
Find the area between \( y = x^2 \) and \( y = x \):
1. Intersection points: \( x = 0 \), \( x = 1 \)
2. \( x \ge 0 \Rightarrow x \ge x^2 \)
3. Area:
\[
\int_0^1 (x - x^2) \, dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}
\]
---
## Examples
[**Video: Solutions**](https://www.youtube.com/watch?v=azvfjIjyRLw)
1. Find the area under the curve of $m(x) = 12 - 0.5x^2$ from $1$ until the point at which the curve hits the $x$-axis.
2. Find the area bounded by $y = 4\sqrt{x}$, $y = 0$ and $x = 4$.
3. Find the area bounded by the curve $y = 11 + 6x + x^2$ and the x-axis, from $x = -3$ to $x = -1$.
4. Sketch the region enclosed by $y = 5x$ and $y = 5x^2$.
5. A sketch of the region enclosed by $y = x$ and $y = \frac{3.25x}{x^2 + 1}$. Find the area.
---
## Practice Problems
1. Find area under $y = e^x$ on $[1,4]$
2. Find area under $y = \sin(x)$ on $[0, \pi]$; net vs true area on $[0, 2\pi]$
3. Area under $y = 1/x^2$ on $[1, \infty]$
4. Why does $\int_1^\infty 1/x dx$ not converge?
5. **(Challenge)** For what $p$ does $\int_1^\infty \frac{1}{x^p} dx$ converge?
6. Find area between $y = x^2$ and $y = x^4$
7. Find area between $y = -|x|$ and $y = x^2 - 2$
8. **(Challenge)** Find net and true area between $y = -x^2 + 4$ and $y = x^2 - 2x$
---
## Self Assessment
Try to solve these in 20 minutes:
1. Area under $y = \ln(x)$ on $[2,8]$
2. Area bounded by $y = x^3$ and the axes in the 4th quadrant
3. Area under $y = e^{-x}$ on $[0,\infty]$
4. Area between $y = -|x|$ and $y = x^2 - 2x$
---
## Lesson Checklist
This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.
\bigskip
\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.}
\end{align*}
| Skill | D | CON | COM |
|-----------------------------------------|---|-----|-----|
| Determine net and true area with $x$-axis. | | | |
| Determine net and true area between curves. | | | |
| Evaluate simple improper integrals. | | | |
---
# Constrained Optimization
## Lecture Content
[**Video: Constrained Optimization**](https://www.youtube.com/watch?v=KRy8ju4jG5E)
---
# Lecture Notes
### Sketching the Feasible Region
To solve a linear programming problem involving maximization, start by identifying the feasible region — the set of all possible solutions that satisfy the constraints.
#### Steps:
1. **Define the variables.**
- Assign \( x \), \( y \), etc., to represent the quantities in the problem.
2. **Write the system of inequalities.**
- These represent the constraints.
3. **Graph each inequality:**
- Convert to an equation to graph the boundary.
- Use a solid line for \( \leq \) or \( \geq \), dashed for \( < \) or \( > \).
- Shade the correct side of each line.
4. **Identify the feasible region:**
- The region where all shaded areas **overlap**.
- It must also satisfy non-negativity constraints (e.g., \( x \ge 0 \), \( y \ge 0 \)).
5. **Label the corner (vertex) points** of the feasible region.
- These are typically found at the intersections of constraint lines.
---
### Maximizing the Objective Function
Once the feasible region is identified and corner points are known, the objective function can be evaluated.
#### Steps:
1. **Write the objective function:**
- Usually in the form \( P = ax + by \), where \( a \), \( b \) are constants.
2. **Plug each corner point** into the objective function to calculate \( P \).
3. **Compare the results:**
- The **largest value** is the **maximum**.
- The **smallest value** is the **minimum**, if needed.
4. **State the optimal solution:**
- Report both the point \( (x, y) \) and the maximum value \( P \).
---
## Example
Maximize \( P = 3x + 5y \), subject to: