2-15-2024: Section 7-4 and Chapter 6 & 7 Review
Hi everyone! Here are the class notes and a review test to help you prepare for the upcoming exam covering Chapters 6 and 7. Remember to practice consistently and don't hesitate to ask questions if you get stuck. Good luck!
Review Test Problems
- Volume by Rotation (Disk/Washer Method): Find the volume of the region enclosed by the curves $y = x$ and $y = x^2$ when rotated around the x-axis. Remember to determine the points of intersection and set up the integral correctly: $$V = \pi \int_a^b (R(x)^2 - r(x)^2) dx$$ where $R(x)$ is the outer radius and $r(x)$ is the inner radius.
- Volume by Rotation (Cylindrical Shells): Determine the volume of the solid obtained by rotating the region bounded by $y = 2x^2 - x^3$ and $y = 0$ about the y-axis. Recall the cylindrical shell method: $$V = 2\pi \int_a^b x f(x) dx$$
- Volume with a Shifted Axis of Rotation: Calculate the volume of the solid formed by rotating the region bounded by $x = y^2$ and $x = 1 - y^2$ around the axis $x = 3$. Think about how the axis shift affects the radius in your integral.
- Indefinite Integral: Evaluate the indefinite integral $\int (3x + 1)\sqrt{2} dx $. Remember the power rule for integration and any necessary substitutions.
- Integration using Trigonometric Identities: Evaluate $\int x \sin(x) \cos(x) dx$. Simplify using the double-angle identity $\sin(2x) = 2\sin(x)\cos(x)$ before integrating.
- Trigonometric Integrals: Evaluate $\int \sin^5(x) \cos^4(x) dx$. Use trigonometric identities to reduce the power of sine and cosine, enabling straightforward integration using u-substitution.
- Trigonometric Substitution: Evaluate $\int \frac{1}{x^2\sqrt{x^2 - 1}} dx$. Recognize that $x = \sec(\theta)$ is a good choice here.
- Partial Fraction Decomposition: Evaluate $\int \frac{2x - 3}{x^3 + 3x} dx$. This will require factoring the denominator and using partial fraction decomposition.
- Partial Fraction Decomposition: Evaluate $\int \frac{x + 2}{x^2 + 3x - 4} dx$. Remember to factor the denominator and set up the partial fraction decomposition correctly.
- Inverse Trigonometric Function: Evaluate $\int \frac{1}{x^3} \cos^{-1}(x) dx$. Integration by parts might be helpful here.
Key Concepts from Section 7-4: Integration of Rational Functions by Partial Fractions
Section 7-4 focuses on integrating rational functions using the method of partial fractions. This technique is essential when dealing with integrals of the form $\int \frac{P(x)}{Q(x)} dx$, where $P(x)$ and $Q(x)$ are polynomials.
Steps for Partial Fraction Decomposition:
- Factor the Denominator: Factor $Q(x)$ completely. The nature of the factors (linear, irreducible quadratic, repeated factors) determines the form of the decomposition.
- Set up the Decomposition: Based on the factors of $Q(x)$, write $\frac{P(x)}{Q(x)}$ as a sum of simpler fractions. Here's a quick reminder of the different cases:
- Distinct Linear Factors: If $Q(x) = (a_1x + b_1)(a_2x + b_2)...(a_kx + b_k)$, then $$\frac{P(x)}{Q(x)} = \frac{A_1}{a_1x + b_1} + \frac{A_2}{a_2x + b_2} + ... + \frac{A_k}{a_kx + b_k}$$
- Repeated Linear Factors: If $(ax + b)^r$ is a factor of $Q(x)$, then include the terms $$\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + ... + \frac{A_r}{(ax + b)^r}$$
- Irreducible Quadratic Factors: If $ax^2 + bx + c$ is a factor of $Q(x)$ (where $b^2 - 4ac < 0$), then include a term of the form $$\frac{Ax + B}{ax^2 + bx + c}$$
- Solve for the Constants: Multiply both sides of the equation by $Q(x)$ and solve for the unknown constants ($A_i$, $B_i$, etc.) by either substituting convenient values of $x$ or equating coefficients of like powers of $x$.
- Integrate: Integrate each of the simpler fractions.
Remember, practice is key! Work through plenty of examples, and you'll master these techniques in no time. You've got this!