Chapter 6 and 7 Review: Ace Your Exam!

Welcome to the Chapter 6 and 7 review page! Here you'll find resources to help you prepare for your upcoming exam. Let's break down the key concepts and ensure you're ready to go!

Key Concepts

  • Area Between Curves: Remember that the area $A$ between two curves $y = f(x)$ and $y = g(x)$, where $f(x) \ge g(x)$ on the interval $[a, b]$, is given by the definite integral: $$A = \int_a^b [f(x) - g(x)] dx$$
  • Volume of Solids of Revolution:
    • Disk/Washer Method: If we rotate a region around an axis, we can find the volume by integrating the area of circular disks or washers.
    • Cylindrical Shells: An alternative method for finding volumes, especially useful when the axis of rotation is parallel to the axis of integration. The volume $V$ is given by $$V = \int_a^b 2\pi x f(x) dx$$ where $f(x)$ is the height of the shell.
  • Integration Techniques: Chapter 7 focuses heavily on integration techniques. Be sure to practice:
    • U-Substitution: Identifying suitable substitutions to simplify integrals.
    • Integration by Parts: Using the formula $\int u dv = uv - \int v du$. Remember LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential) to help choose $u$.
    • Trigonometric Integrals: Mastering integrals involving trigonometric functions using trigonometric identities. For example, using $\sin^2(x) + \cos^2(x) = 1$ and half-angle formulas.
    • Trigonometric Substitution: Using trigonometric substitutions to evaluate integrals involving radicals like $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, and $\sqrt{x^2 - a^2}$.
    • Partial Fractions: Decomposing rational functions into simpler fractions for easier integration.

Important Formulas

Here's a quick recap of some essential integration formulas:

  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
  • $\int \frac{1}{x} dx = \ln |x| + C$
  • $\int e^x dx = e^x + C$
  • $\int \sin x dx = -\cos x + C$
  • $\int \cos x dx = \sin x + C$
  • $\int \sec^2 x dx = \tan x + C$

Resources

Keep practicing, and you'll do great on the exam! Remember to review your notes, work through practice problems, and don't hesitate to ask questions. Good luck, Professor Baker's Math Class!