Student Guide:The final integration technique we cover is Partial Fractions in Section 7.4. This is a fairly complex algebraic technique for simplifying rational functions. To make things easier, we cover only the cases where the denominator factors into linear terms, is itself an irreducible quadratic, or factors into (possibly repeated) linear terms times a single irreducible quadratic term. This should give you the basic idea of the technique. You'll probably have to review long division of polynomials. The technique of Rationalizing Substitutions is also in this section.Integrals requiring several techniques are presented in Section 7.5. These can get rather difficult. The homework focuses on the more straight-forward examples. We have already seen how to approximate integrals using Riemann Sums. This works quite well if we use midpoints for our sample points. Section 7.7 introduces two more techniques for approximating integrals that you just can't compute, the Trapezoid Rule and Simpson's Rule. These are key techniques for use in applications. Most instructors skip the error estimation formulas. They are mostly of theoretical interest. In practical applications, the error of your approximation is estimated using other methods.
|