Welcome to Calculus 2 - Spring Semester!

Hello everyone, and welcome to Calculus 2! I'm excited to embark on this mathematical journey with you. This post provides all the essential information you need to get started in our Spring 2024 Calc 2 class. Please read carefully and don't hesitate to reach out if you have any questions.

Course Information

Class Syllabus: Access the course syllabus for a comprehensive overview of the course structure, grading policies, and important dates.

Homework: All homework assignments will be submitted through WebAssign. Be sure to register as soon as possible!

WebAssign Course Code: TRCC.MOHEGAN52400071

Online Class Participation: We will be using Google Meet for online class sessions. If you plan to join remotely, please email me to let me know so I can ensure you receive all necessary materials and instructions.

Class Notes: Lecture notes will be posted regularly after each class. Check back here often!

Section 6.1: Area Between Curves (Key Concepts)

Our first major topic is finding the area between curves. This builds directly on your knowledge of definite integrals from Calculus 1. Remember that a definite integral, $\int_a^b f(x) dx$, represents the area under the curve $f(x)$ from $x=a$ to $x=b$.

To find the area between two curves, $f(x)$ and $g(x)$, where $f(x) \ge g(x)$ on the interval $[a, b]$, we use the following formula:

$$Area = \int_a^b [f(x) - g(x)] dx$$

Key Idea: We're integrating the difference between the 'top' function, $f(x)$, and the 'bottom' function, $g(x)$.

Important Considerations:

  • Intersection Points: You'll often need to find the points where the curves intersect. These points will define your limits of integration ($a$ and $b$). Set $f(x) = g(x)$ and solve for $x$.
  • Which function is on top?: The function with the larger $y$-value is the 'top' function. If the functions switch positions over the interval, you'll need to split the integral into multiple integrals.
  • Integrating with respect to $y$: Sometimes it's easier to integrate with respect to $y$. In this case, you'll need to express your functions as $x = f(y)$ and $x = g(y)$, and integrate the difference between the 'right' function and the 'left' function. The limits of integration will be $y$-values.

Example: Find the area between the curves $y = x^2$ and $y = 2x$ from $x = 0$ to $x = 2$.

Here, $2x \ge x^2$ on the interval $[0, 2]$. So, the area is given by:

$$Area = \int_0^2 (2x - x^2) dx = \left[x^2 - \frac{x^3}{3}\right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}$$

Remember to practice these concepts through the WebAssign homework. Good luck, and I look forward to a productive semester!