Final Exam Review - Let's Get Ready!

Welcome to your final exam review! This page contains helpful resources to guide you through the material and prepare you for success. Remember to stay focused, practice consistently, and don't hesitate to ask questions. You've got this!

Key Concepts and Practice Problems

Based on the provided materials, here are some crucial concepts and examples to review:

  • Volume of Solids of Revolution: Understanding how to calculate volumes using integration, especially with the disk/washer method and the shell method.

Example (Disk Method): Find the volume $V$ of the solid obtained by rotating the region bounded by $y = x + 1$, $y = 0$, $x = 0$, and $x = 2$ about the x-axis.

Here, $A(x) = \pi r^2 = \pi (x+1)^2 = \pi (x^2 + 2x + 1)$.

Then, the volume $V$ is given by:

$$V = \int_0^2 \pi (x^2 + 2x + 1) dx = \pi \left[ \frac{x^3}{3} + x^2 + x \right]_0^2 = \pi \left( \frac{8}{3} + 4 + 2 \right) = \frac{26\pi}{3}$$
  • Shell Method: A method for finding the volume of a solid of revolution when integrating parallel to the axis of revolution.

Example (Shell Method): The volume of the solid obtained by rotating about the y-axis the region under $y=f(x)$ from $a$ to $b$ is $V = \int_a^b 2\pi x f(x) dx$

  • Integration Techniques: Mastering various integration techniques, including trigonometric substitution and partial fractions.

Example (Trigonometric Substitution): Evaluating integrals involving trigonometric functions, such as $\int \sin(t) \cos(t) dt$. Using the identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, we can rewrite the integral as:

$$\int x \cdot (2\sin(x) \cos(x)) dx = \int x \sin(2x) dx$$

Applying integration by parts, let $u = x$ and $dv = \sin(2x) dx$. Then, $du = dx$ and $v = -\frac{1}{2}\cos(2x)$.

So, the integral becomes:

$$x(-\frac{1}{2}\cos(2x)) - \int -\frac{1}{2}\cos(2x) dx = -\frac{1}{2}x\cos(2x) + \frac{1}{2} \int \cos(2x) dx = -\frac{1}{2}x\cos(2x) + \frac{1}{4}\sin(2x) + C$$
  • Improper Integrals and Convergence/Divergence: Using the Integral Test to determine convergence/divergence of series.

Example (Integral Test): Determine whether $\sum_{n=1}^{\infty} \frac{10}{n^{1.5}}$ converges or diverges by evaluating $\int_1^{\infty} \frac{10}{x^{1.5}} dx $.

  • Power Series: Finding power series representations for functions and determining their radius of convergence.

Example (Power Series): Finding a power series representation for $f(x) = \frac{x}{(1+4x)^2}$. Remember the geometric series formula: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$

  • Vectors: Vector operations, dot product, cross product, and finding angles between vectors.

Example (Vector Operations): If $a = <9, -8, 7>$ and $b = <7, 0, -9>$

Then $a+b = <16,-8,-2>$. And $|a| = \sqrt{9^2+(-8)^2+7^2} = \sqrt{194}$

Final Thoughts

Keep practicing, stay confident, and review these notes carefully. Good luck on your final exam! Professor Baker believes in you!