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Chapter 6: Q. 46 (page 511)

Consider the region between the graphs of $f (x) = 5 - x$ and $g (x) = 2 x$ on $[1, 4]$ For each line of rotation given in Exercises 45 and 46, use definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The volume of the solid is $\frac{1894}{27} 蟺$

Step by step solution

01

Step 1. Given Information

The given figure is

$f (x) = g (x) 5 - x = 2 x x = \frac{5}{3}$

02

Step 2. Finding Volume

$V_{1} = 蟺 {鈭�}_{1}^{\frac{5}{3}} {(5 - 1)}^{2} - {(2 x - 1)}^{2} dx V_{1} = 蟺 {鈭�}_{1}^{\frac{5}{3}} (16 - 4 x^{2} - 1 + 4 x) dx = 蟺 {鈭�}_{1}^{\frac{5}{3}} (15 - 4 x + 3 x^{2}) dx V_{1} = 蟺 {[15 x - 2 x^{2} + x^{3}]}_{1}^{\frac{5}{3}} = 蟺 [\frac{75}{3} - \frac{50}{9} + \frac{125}{27} - (15 - 2 + 1)] = 蟺 (\frac{552}{27} - 14) = \frac{174}{27} 蟺 = \frac{58}{9} 蟺$

$V_{2} = 蟺 {鈭�}_{\frac{5}{3}}^{4} (3 x^{2} + 4 x - 15) dx V_{2} = 蟺 {[x^{3} + 2 x^{2} - 15 x]}_{\frac{5}{/ 3}}^{4} = 蟺 [64 + 32 - 60 - (\frac{125}{27} + \frac{50}{9} - \frac{75}{3})] = 蟺 (36 + \frac{400}{27}) = \frac{1372}{27} 蟺$

03

Step 3. Finding Final Volume

$V = V_{1} + V_{2} V = \frac{58}{9} 蟺 + \frac{1372}{27} 蟺 = \frac{1894}{27} 蟺$

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Most popular questions from this chapter

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$蟺 {鈭�}_{0}^{\frac{蟺}{2}} \cos^{2} (x) d x a n d 蟺 {鈭�}_{\frac{蟺}{2}}^{蟺} \cos^{2} (x) d x$

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29-52.

$\frac{d y}{d x} = y^{2} \cos x, y (蟺) = 1$

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52

$\frac{d y}{d x} = \sqrt{4 y}, y (1) = 1$

Suppose a population P(t) of animals on a small island grows according to a logistic model of the form $\frac{d P}{d t} = k P (1 - \frac{P}{100})$ for some constant $K$ .

(a) What is the carrying capacity of the island under this model?

(b) Given that the population $P (t)$ is growing and that $0 < P (t) < 500$ , is the constant k positive or negative, and why?

(c) Explain why $\frac{d P}{d t} 鈮� k P$ for small values of $P$ .

(d) Explain why $\frac{d P}{d t} 鈮� 0$ for values of $P$ that are close to the carrying capacity

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