Chapter 8 and 9 Review Test - Prepare to Excel!

Hello Professor Baker's Math Class! This post is designed to help you prepare for the upcoming Chapter 8 and 9 review test. We'll cover key concepts and provide examples to boost your confidence. Let's get started!

Key Topics

  • Arc Length: Calculating the length of a curve over a given interval.
  • Surface Area of Revolution: Finding the area of a surface generated by rotating a curve around an axis.
  • Separable Differential Equations: Solving differential equations by separating variables.

Arc Length

The arc length of a function $y = f(x)$ from $x = a$ to $x = b$ is given by the formula:

$$L = \int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^2} dx$$

Example Problems:

  1. Find the arc length of $y = \frac{2}{3}x^{\frac{3}{2}}$, $0 \le x \le 2$.
  2. Find the arc length of $y = \frac{x^4}{8} + \frac{1}{4x^2}$, $1 \le x \le 2$.
  3. Find the arc length of $y = ln(sin(x))$, $\frac{\pi}{3} \le x \le \frac{\pi}{2}$.

Surface Area of Revolution

The surface area of revolution when the curve $y = f(x)$, $a \le x \le b$, is rotated about the x-axis is given by:

$$S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + (\frac{dy}{dx})^2} dx$$

When rotated about the y-axis, the surface area is:

$$S = 2\pi \int_{a}^{b} x \sqrt{1 + (\frac{dy}{dx})^2} dx$$

Example Problems:

  1. What's the area of the surface of revolution created when the curve defined by $y = 4 + 3x^2$, $1 \le x \le 2$ is rotated around the y-axis?
  2. What's the area of the surface of revolution created when the curve defined by $x = cos^2(y)$, $0 \le y \le \frac{\pi}{2}$ is rotated around the y-axis?
  3. What's the area of the surface of revolution created when the curve defined by $y = x \cdot ln(x)$, $1 \le x \le 2$ is rotated around the x-axis?

Separable Differential Equations

A separable differential equation can be written in the form:

$$\frac{dy}{dx} = g(x)f(y)$$

To solve, separate the variables and integrate:

$$\int \frac{1}{f(y)} dy = \int g(x) dx$$

Example Problems:

  1. Solve the equation: $\frac{dy}{dx} = \frac{x^2 - 1}{2y^2}$
  2. Solve the equation: $x + 3y^2\sqrt{x^2 + 1} \frac{dy}{dx} = 0$, $y(0) = 1$

Example: Solve the differential equation $\frac{dy}{dx} = \frac{x^2}{y^2}$ with the initial condition $y(0) = 2$.

Separate variables and integrate: $\int y^2 dy = \int x^2 dx$ which gives $\frac{y^3}{3} = \frac{x^3}{3} + C$.

Then $y^3 = x^3 + 3C$. Applying the initial condition: $2^3 = 0^3 + 3C$, so $C = \frac{8}{3}$.

The solution is $y = \sqrt[3]{x^3 + 8}$

Mixture Problems

A common application of differential equations involves mixture problems. These problems often involve a tank with a solution entering and leaving at certain rates.

Example Problem: A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?

Revenue Problems

Another application deals with revenue flow.

Example Problem: If revenue flows into a company at a rate of $f(t) = 9000\sqrt{1 + 2t}$, where $t$ is measured in years and $f(t)$ is measured in dollars per year, find the total revenue obtained in the first four years.

Good luck with your test! Remember to review your notes, practice these problems, and ask questions if you're unsure about anything. You've got this!