Chapter 10 Projects: Parametric Equations and Polar Curves
Hello Math Mavericks! For Chapter 10, we're ditching the traditional test and diving into two exciting projects. These projects are designed to reinforce your understanding of parametric equations and polar curves in a fun and interactive way. You're welcome to use any online tools to help you with your work.
Key Dates:
- Due Date: April 13th
- Recommendation: Since we're moving on to Chapter 11 next week, I recommend completing these projects this week to stay on track.
Project Details:
Project 1: Section 10.2 - You Gotta Have Heart (Parametric Equations)
This project focuses on parametric equations. You will be working with the parametric curve defined by:
$$x = (cos(t))(1 + cos(t)), y = (sin(t))(1 + cos(t)), 0 \le t \le 2\pi$$Here's what you'll need to do:
- Graph the Curve: Carefully plot the curve defined by the parametric equations. Think about the range of $t$ and how it affects the shape.
- Surface Area Integral: Set up an integral to find the surface area formed by rotating the portion of the curve in the first quadrant about the x-axis. Remember the surface area formula for parametric curves!
- Simplify and Compute: Show that your integral can be simplified to: $$\int_0^{\pi/2} 2\sqrt{2}\pi (1 + cos(t))^{3/2} sin(t) dt$$ and then compute the integral to find the surface area. This will require a u-substitution.
- Explain the Title: Why do you think this project is called "You Gotta Have Heart"? (Hint: Consider the shape of the curve!)
Project 2: Section 10.4 - Polar Propellers (Polar Curves)
In this project, you'll be exploring polar curves. You'll be working with the polar curve defined by:
$$r = 2 + cos(4\theta), 0 \le \theta \le 2\pi$$Here's what you'll need to do:
- Graph and Farthest Points: Graph this curve in polar coordinates. Determine the points that are farthest from the origin and the corresponding values of $\theta$. Think about when the cosine function reaches its maximum value.
- Closest Points: Determine the points that are closest to the origin and the corresponding values of $\theta$. Consider when the cosine function reaches its minimum value.
- Area Bounded: Find the area bounded by this polar curve. Remember the formula for area in polar coordinates involves an integral.
- Arc Length Integral: Set up the integral for the arc length of this polar curve. Don't forget the formula involves derivatives!
Important Note: These projects are designed to be collaborative and hands-on. Don't hesitate to ask questions and work together to solve the problems.
Chapter 11 will consist of an in-class test, as the material is better suited to that assessment format. Good luck, and have fun exploring these fascinating topics!