Welcome back, students! In our latest session, we conducted a thorough review for the Chapter 6-7 Test. These chapters cover some of the most visually interesting and mechanically challenging topics in Calculus II: Applications of Integration and Techniques of Integration. If you missed class or just need a refresher, this post breaks down the key problems we solved in the lecture notes.
1. Volumes of Revolution: The Washer Method
We started by finding the volume of a region bounded by $x = y^2$ and $x = 1-y^2$ rotated about the vertical line $x = -1$.
Key Strategy: Since we are rotating around a vertical line and our functions are given as $x=f(y)$, it is most efficient to integrate with respect to $y$. Because there is a gap between the axis of rotation and the region, we must use the Washer Method.
The general formula for the Washer Method is:
$$V = \pi \int_a^b ([R(y)]^2 - [r(y)]^2) \, dy$$In our specific example (Problem 1), the outer radius $R$ extends from the axis $x=-1$ to the far curve, and the inner radius $r$ extends to the near curve:
- Right Curve (Outer): $1 + (1-y^2) = 2-y^2$
- Left Curve (Inner): $1 + y^2$
Don't forget to find your intersection points to set your limits of integration!
2. The Integration Hierarchy
Chapter 7 is all about having a toolkit of integration strategies. Before diving into complex methods, always follow this mental checklist (as seen in the notes):
- Simplify: Can algebraic manipulation or trig identities make the integral easier immediately?
- u-Substitution: Look for a function $g(x)$ and its derivative $g'(x)$ inside the integral.
- Integration by Parts: Best for products of functions (like $x^4 \ln x$).
- Trig Integrals & Substitution: Used for powers of trig functions or radicals like $\sqrt{x^2-a^2}$.
- Partial Fractions: Used for rational functions.
3. Spotlight on Tricky Integrals
Integration by Parts
For Problem 4, we looked at $\int x^4 \ln x \, dx$. Remember the acronym LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential) to choose your $u$. Here, $u = \ln x$ is the correct choice because we can easily differentiate it to get $\frac{1}{x}dx$, while integrating $\ln x$ is more complex.
$$ \int u \, dv = uv - \int v \, du $$The "Boomerang" Integral
In Problem 5 ($\int e^{3x} \cos x \, dx$), we encountered a classic scenario where integrating by parts twice leads you back to the original integral. Don't panic! Simply treat the integral as an algebraic variable, move it to the other side of the equation, and solve for it.
Partial Fractions with Long Division
Problem 9 presented a common trap: $\int_1^2 \frac{3x^2+6x+2}{x^2+3x+2} \, dx$.
Important Rule: If the degree of the numerator is greater than or equal to the degree of the denominator, you must perform Polynomial Long Division first. Only apply Partial Fraction Decomposition to the remainder.
4. Trigonometric Substitution
Finally, we reviewed radicals. When you see a form like $\sqrt{x^2 - 16}$, think of the Pythagorean identities. By substituting $x = 4 \sec \theta$, we transformed the radical into $4 \tan \theta$, simplifying the integral significantly.
Study hard, review these examples, and remember: identifying the correct technique is half the battle. Good luck on the test!