Exam #2: Review Exercises + Example Videos

As we’ve been discussing in class, our 2nd exam will be this Wednesday, April 15. Below is the outline of topics that will be covered on the exam. The HW#8 exercises cover these topics:

  • From the Final Exam Review sheet (handed out in class, also available here):
    • #5 (Sec 7.3: Trigonometric Substitutions)
    • #6 (Sec 7.5, Method of Partial Fractions)
    • #7 (Sec 7.6, Improper Integrals)
  • Sec 6.1, Integrals for the area between two curves: (pp361-362): #1, 3, 4, 27
  • Sec 6.3, Volumes of Revolution (p381): #1, 3, 5, 6

Trigonometric Substitutions:

Do #5(a), (b), (c), and (d) on the Final Exam Review Sheet. Take a look at the outline of how to approach these problems I posted in the discussion forum.

Actually we did #5(a) in class today. If you want to review it again (or missed class today), here is a video I did of a very similar example:

If you want to review this topic further, look at Examples 1, 2 & 4 in Section 7.3. You could also review #53 from Sec 7.5, which was assigned as a HW exercise.


Method of Partial Fractions

There are three types of partial fractions integrands we discussed, depending on the factors that appear in the denominator:

  1. “Distinct linear factors”: This is the most basic type, where the partial fraction decomposition has a constant coefficient for each distinct linear factor. For example, in Example 2 in Section 7.5, the partial fraction decomposition of the integrand is of the form:\frac{x^2 + 2}{(x-1)(2x-8)(x+2)} = \frac{A}{x-1} + \frac{B}{2x-8} + \frac{C}{x+2}You should be familiar with how to solve for A, B and C in an example like this. If you need to review, look at Examples 1 and 2 in Section 7.5.#6(b) and #6(c) on the Final Exam Review Sheet are integrands of this type (after you factor the given denominators!)
  2. “Repeated linear factors”: Here the partial fraction decomposition must include terms with constant coefficients for each power of the repeated linear factor.For example, consider #17 from Section 7.5, where the integrand is\frac{x^2 + 11x}{(x-1)(x+1)^2}.Since the (x+1) factor is repeated twice, the partial fraction decomposition must be of the form:\frac{x^2 + 11x}{(x-1)(x+1)^2} = = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{(x+1)^2}I worked out the rest of this exercises here:

    If you need to review further, look at Examples 4 and 5(c) in Section 7.5.

  3. “Irreducible quadratic facors”: If the denominator contains an irreducible quadratic factor, i.e., a quadratic term (such as x^2 + 9) that can’t be factored into linear factors. In that case we include a term in the partial fraction decomposition for such a quadratic factor, but the numerator isn’t just a constant but must be a linear term.This is the case with #6(a) on the Final Exam Review. Review in the example we did in class today, and also read Example 5(a) in Sec 7.5 carefully. You can also view this video by patrickJMT (who has a bunch of integration examples that you may find helpful!)

Improper Integrals

Do the three improper integrals in #7 of the Final Exam Review sheet. For each, first rewrite the integrals as a limit, and then evaluate the definite integral inside the limit–for each of these three, you can find the antiderivative by making a simple u-substitution.   Finally evaluate the limit

Check back here later, I’ll post an example in case you need some guidance with these.

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