Welcome to the final stretch of the Spring 2024 semester! As we approach the Calculus 2 Final Exam, it is crucial to consolidate everything we have learned, from complex integration methods to the convergence of infinite series. To help you prepare effectively, I have compiled a comprehensive Final Exam Review sheet along with detailed solutions.

Below, you will find a breakdown of the key topics covered in the review, along with the study materials you need to succeed.

Part 1: Integration Techniques (Problems 1-5)

The first section of the exam focuses on your ability to recognize which integration technique applies to a given function. The review covers:

  • U-Substitution: Handling composite functions like $\int_0^1 \frac{3x}{(2x+1)^3} dx$.
  • Trigonometric Integrals: Using identities to solve integrals involving powers of sine, cosine, tangent, and secant, such as $\int_0^{\pi/4} \tan^3(\theta)\sec^2(\theta) d\theta$.
  • Partial Fractions: Decomposing rational functions where the denominator contains irreducible quadratic factors, specifically looking at $\int \frac{2x-3}{x^3+3x} dx$.
  • Trigonometric Substitution: Recognizing forms like $\sqrt{4x+49}$ (which relates to $\sqrt{x^2+a^2}$) to simplify the integrand.

Part 2: Applications of the Integral (Problems 6-10)

Calculus is not just about finding the area under a curve; it is about applying that concept to 3D space and real-world rates of change. Be prepared to compute:

  • Volumes of Revolution: You must decide between the Disk/Washer Method ($V = \pi \int [R(x)^2 - r(x)^2] dx$) and the Shell Method ($V = 2\pi \int x f(x) dx$). Review Problems 6 and 7 carefully to see the difference between rotating around the x-axis versus the y-axis.
  • Arc Length and Surface Area: utilizing the formula $L = \int \sqrt{1 + [f'(x)]^2} dx$.
  • Accumulation of Change: Problem 10 presents a real-world scenario regarding a mosquito population increasing at a rate of $n(t) = 1500 + 10e^{0.9t}$. Remember that the net change in population is found by integrating the rate function over the given time interval.

Part 3: Infinite Series (Problems 11-15)

The final portion of the course is often the most challenging. We transition from continuous functions to discrete series. The review emphasizes:

  • Convergence Tests: You need to be proficient with the Ratio Test (especially for factorials like $\sum (-1)^n \frac{\pi^{4n}}{(2n)!}$) and Comparison Tests.
  • Power Series: Finding the Radius and Interval of Convergence. For Problem 14, $\sum_{n=1}^{\infty} \frac{(-1)^n 3^n}{\sqrt{n}} x^n$, remember to check the endpoints of your interval!
  • Maclaurin Series Applications: Using known series expansions (like $\cos(x)$) to evaluate non-elementary integrals as infinite series.

Study Resources

Review the PDF documents linked below. I highly recommend attempting the blank review sheet first before checking your work against the handwritten solutions. Watching the video lectures will also provide context on why we chose specific methods for each problem.

Good luck studying! You have done the hard work all semester—finish strong!