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Chapter 6: Q. 1 (page 573)
Fill in the blanks to complete each of the following theorem statements.
The volume of a shell with height , average radius , and thickness is .
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The volume of solid obtained by revolving the region between the graph around (a)the y axis (b)the line x=2
Prove Theorem 6.22 by solving the initial-value problem with T(0) = T0, where k and A are constants.
Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52
Suppose a population P(t) of animals on a small island grows according to a logistic model of the form for some constant .
(a) What is the carrying capacity of the island under this model?
(b) Given that the population is growing and that , is the constant k positive or negative, and why?
(c) Explain why for small values of .
(d) Explain why for values of that are close to the carrying capacity
Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.
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