Chapter 6 and 7 Review Test

Here's a review test to help you prepare for your upcoming exam covering Chapters 6 and 7. Remember to review your notes, practice problems, and understand the key concepts we've covered in class. Good luck, and don't hesitate to ask questions if you're stuck! This review test will help you solidify your understanding of integration and its applications.

Practice Problems

  1. Volume of Revolution: Find the volume of the region enclosed by the curves $y = x$ and $y = x^2$ when it is rotated around the x-axis. Remember to use the disk or washer method: $$V = \pi \int_a^b (R(x)^2 - r(x)^2) dx$$, where $R(x)$ is the outer radius and $r(x)$ is the inner radius.
  2. Integration: Complete the integration of the following: $\int (3x + 1)^{\sqrt{2}} dx$. Consider using u-substitution. Let $u = 3x+1$. Remember to adjust your $du$ accordingly!
  3. Trigonometric Integration: Evaluate $\int x \sin(x) \cos(x) dx$. Hint: Use the double angle identity $\sin(2x) = 2\sin(x)\cos(x)$ to simplify the integrand. Then, try integration by parts.
  4. Trigonometric Integrals: Determine $\int \sin^5(x) \cos^4(x) dx$. Consider using u-substitution and trigonometric identities. If the power of sine is odd, save one sine and convert the rest to cosine.
  5. Trigonometric Substitution: Solve $\int \frac{1}{x^2 \sqrt{x^2 - 1}} dx$. Use the substitution $x = \sec(\theta)$. Remember that $\sqrt{\sec^2(\theta) - 1} = \tan(\theta)$.
  6. Partial Fractions: Compute $\int \frac{2x - 3}{x^3 + 3x} dx$. First, factor the denominator. Then, decompose the rational function into simpler fractions. Remember to solve for the coefficients.
  7. Exponential Integration: Evaluate $\int \frac{1}{1 + e^x} dx$. Try multiplying the numerator and denominator by $e^{-x}$.
  8. Partial Fractions: Determine $\int \frac{x+2}{x^2+3x-4} dx$. Factor the denominator and use partial fraction decomposition.
  9. Inverse Trig Functions: Evaluate $\int \frac{\cos^{-1}(x)}{x^3} dx$. Try integration by parts.
  10. Improper Integrals: Determine $\int_3^{\infty} \frac{1}{(x-2)^2} dx$. Remember to rewrite the integral as a limit and evaluate. $$ \lim_{b \to \infty} \int_3^b \frac{1}{(x-2)^2} dx $$.

Key Concepts to Review:

  • Volumes of Revolution: Disk method, washer method, and shell method.
  • Integration Techniques: u-substitution, integration by parts, trigonometric integrals, trigonometric substitution, partial fractions.
  • Improper Integrals: Evaluating integrals with infinite limits or discontinuous integrands.

Remember to show all your work and check your answers. Good luck with your test!