Chapter 6: Q. 47 (page 512)
Consider the region between the graphs ofand on .For each line of rotation given in Exercises 47–50, use definite integrals to find the volume of the resulting solid.
The volume of the solid is
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Given an initial-value problem, we can apply Euler’s method to generate a sequence of points , and so on. How are these coordinate points related to the solution of the initial-value problem?
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
Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52.
37.
Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .
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