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Section 4.6

By cssp

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) \displaystyle{\int_1^\infty \frac{1}{x^3} \ dx = \frac{1}{2}}

(b) \displaystyle{\int_4^\infty \frac{3}{x^7} \ dx = \frac{1}{8192}}

(c) \displaystyle{\int_{10}^\infty \frac{1}{5x^4} \ dx = \frac{1}{15000}}

(d) This integral diverges.

(e) \displaystyle{\int_0^\infty \frac{3}{2x + 3)^2} \ dx = \frac{1}{2}}

(f) This integral diverges.

2. (a) \displaystyle{\int_{-\infty}^{-2} \frac{3}{x^2} \ dx = \frac{3}{2}}

(b) \displaystyle{\int_{-\infty}^\infty \frac{x}{(x^2 + 4)^4} \ dx = 0}

(c) This integral diverges.

(d) This integral diverges.

3. (a) Converges

(b) Converges

(c) Diverges

(d) Converges

(e) Converges

(f) Converges

4. (a) \displaystyle{\int_0^8 \frac{1}{x^{\frac{1}{3}}} \ dx = 6}

(b) This integral diverges.

(c) \displaystyle{\int_0^1 \frac{1}{\sqrt{1-x}} \ dx = 2}

(d) \displaystyle{\int_0^5 \frac{5}{(t - 2)^{\frac{2}{5}}} \ dt = \frac{25}{3}\left(2^{\frac{3}{5}} + 3^{\frac{3}{5}}\right)}

(e) This integral diverges.

(f) \displaystyle{\int_{-1}^2 \frac{3}{x^\frac{1}{3}} \ dx = \frac{9}{2}\left(2^\frac{2}{3} - 1\right)}

5. (a) \displaystyle{\int_1^\infty \frac{1}{x^p} \ dx = \frac{1}{p - 1}}

6. (a) \displaystyle{\int_0^1 \frac{1}{x^p} \ dx = \frac{1}{1 - p}}

8. (a) \displaystyle{P(x) = \left(\frac{\sigma}{x}\right)^\alpha}

(b) \displaystyle{A = \frac{\alpha}{\alpha - 1}\sigma}

(d) A = 60,000 , P(A) = 0.1165 , P(2A) = 0.0507
These values tell us that only 11.65 % of the population have incomes exceeding the average income of 60,000 and only 5.07 % of the population have incomes exceeding 120,000 .

(e) \displaystyle{P(A) = \left(\frac{\alpha - 1}{\alpha}\right)^\alpha}, \alpha > 1

This entry was posted on 24 July 24 2007 at 10:13 am and is filed under Section 4.6. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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