Welcome back, class! Today’s session is a crucial checkpoint in our semester. We are diving deep into the Chapter 8 and 9 Test Review. These chapters represent a transition from standard integration techniques to applying those integrals in geometry (Arc Length and Surface Area) and then shifting gears into the world of Differential Equations.

The attached review sheet contains 10 problems that mirror the difficulty and concepts you can expect on the upcoming exam. Let’s break down the key themes found in this practice set.

1. Arc Length (Problems 1-3)

The first few problems focus on finding the length of a curve over an interval. Remember the fundamental formula for arc length $L$ from $a$ to $b$:

$$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$
  • Problem 1: You are given $x$ in terms of $y$. Don’t forget that you will need to differentiate with respect to $y$, so your integral will be $dy$. The algebra here usually involves a perfect square under the radical.
  • Problem 2: A classic trigonometric logarithm problem: $y = \ln(\cos x)$. Recall your trig identities, specifically that $1 + \tan^2(x) = \sec^2(x)$, to simplify the radical.
  • Problem 3 (Application): This is a great engineering problem! You need to calculate the width of a flat metal sheet required to create a corrugated sine wave panel. The curve is $y = \sin(\frac{\pi x}{7})$. This is a direct application of arc length in manufacturing.

2. Surface Area of Revolution (Problems 4-7)

Next, we move to 3D geometry by rotating curves around an axis. The formula for Surface Area $S$ is:

$$S = 2\pi \int_a^b r(x) \sqrt{1 + [f'(x)]^2} \, dx$$

Crucial Tip: Pay close attention to the radius of rotation, $r(x)$.

  • If rotating around the x-axis (like in Problems 4, 5, and 7), the radius is usually the function height $y$.
  • If rotating around the y-axis (like in Problem 6), the radius is the horizontal distance $x$.
  • Problem 7: You are asked to find the surface area of an ellipsoid formed by rotating $\frac{x^2}{9} + \frac{y^2}{4} = 1$. You will need to solve for $y$ (or use implicit differentiation) to set up your integral.

3. Differential Equations (Problems 8-9)

Chapter 9 introduces us to Differential Equations. The main strategy we are reviewing here is Separation of Variables.

For example, in Problem 8 ($xyy' = x^2 + 1$), your goal is to get all terms involving $y$ on one side and all terms involving $x$ on the other before integrating both sides:

$$y \, dy = \frac{x^2+1}{x} \, dx$$

Problem 9 is an Initial Value Problem (IVP). Once you find the general solution with $+C$, use the condition $x(0) = -5$ to solve for the specific constant.

4. Modeling with Differential Equations (Problem 10)

Finally, we have the "Mixing Problem." This is a classic rate problem involving $CO_2$ concentration in a room.

  • Concept: $\frac{dy}{dt} = \text{Rate In} - \text{Rate Out}$
  • Setup: The room volume is $200 \, m^3$. Fresh air flows in at $4 \, m^3/min$ and leaves at the same rate.
  • Watch your units! The rate is given in minutes, but the final question asks for the percentage after 2 hours. Ensure you evaluate your function at $t=120$.

Download the PDF notes attached to this post to see the full problem statements. Work through these carefully—if you can master these 10 problems, you are in excellent shape for the test!