We review substitution and the use of integral tables.
Note: The section on the Gompertz Model (7:08–11:31) relates to ideas that we will study later but have not yet seen. It is recommended that you do your best to understand it now and then come back to it again when we study ODEs.
- OpenStax II 1.5: Substitution and OpenStax II 3.5: Tables and Computer Systems
- Ximera OSU: Substitution
- Community Calculus 8.1: Substitution
If the variable is used for slicing, then slices are perpendicular to the axis of rotation, which indicates the washer method should be used. The inequalities for give the outer and inner radii, and (Note that the absolute values go away when the radius is squared.) This leads to the conclusion Among the options below, the best choice for a potential substitution is
- When dealing with quadratic expressions such as the ones appearing in this integrand, it is often necessary to complete the square before appealing to a table. In this case,
- Using the most appropriate entry of the table and plugging in the correct value of gives
- Based on the results of completing the square, we make a substitution and conclude
- The key is to make the substitution so that the expression can be understood as a quadratic function of .
- Specifically , so Aside from the factor of , the integral belongs to the table:
- We conclude that Absolute values are not needed in the logarithm because can never be negative.