Q. 22 Complete Example 2 by showing th... [FREE SOLUTION]

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Chapter 13: Q. 22 (page 1038)

Complete Example 2 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} \frac{k}{3} (8 \cos^{3} 胃 - \sec^{3} 胃) d 胃 = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

Short Answer

Expert verified

The mass of semicircular lamina is

$m = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

Step by step solution

Given information

The expression is

{鈭�}_{- 蟺 / 4}^{蟺 / 4} \frac{k}{3} (8 \cos^{3} 胃 - \sec^{3} 胃) d 胃 = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))

Calculation

Density $蚁 (r) = k r$

Equation of circle

$r = 2 \cos 胃$

Equation of line $x = 1$ in polar form $r = \sec 胃$

Mass of the plate

$m = {鈭�}_{惟} 蚁 (x, y) d A m = {鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} k r^{2} d r d 胃$

Integrate the inner integral with respect to $r$ first.

$m = k {鈭�}_{- 蟺 / 4}^{蟺 / 4} {[\frac{r^{3}}{3}]}_{\sec 胃}^{2 \cos 胃} d 胃 m = k {鈭�}_{- n / 4}^{蟺 / 4} [\frac{8 \cos^{3} 胃 - \sec^{3} 胃}{3}] d 胃$

Substitute the limits

$m = \frac{k}{3} {鈭�}_{- 蟺 / 4}^{蟺 / 4} [8 \cos^{3} 胃 - \sec^{3} 胃] d 胃$

$m = \frac{k}{3} [\frac{8}{12} {9 \sin 胃 + \sin 3 胃} - \frac{1}{2} {\{\begin{array}{l} \tan 胃 \sec 胃 - \ln (\cos (\frac{胃}{2}) - \sin (\frac{胃}{2})) + \\ \ln (\cos (\frac{胃}{2}) + \sin (\frac{胃}{2})) \end{array}]}_{- 蟺 / 4}^{蟺 / 4}$

$m = \frac{k}{3} [\begin{array}{l} \frac{8}{12} \{9 \sin (\frac{蟺}{4}) + \sin (\frac{3 蟺}{4})\} - \frac{1}{2} \{\begin{array}{l} \tan (\frac{蟺}{4}) \sec (\frac{蟺}{4}) - \ln (\cos (\frac{蟺}{8}) - \sin (\frac{蟺}{8})) + \\ \ln (\cos (\frac{蟺}{8}) + \sin (\frac{蟺}{8})) d \end{array}\} \\ - \frac{8}{12} \{9 \sin (- \frac{蟺}{4}) + \sin (- \frac{3 蟺}{4})\} + \frac{1}{2} \{\begin{array}{l} \tan (- \frac{蟺}{4}) \sec (- \frac{蟺}{4}) - \ln (\begin{array}{l} \cos (- \frac{蟺}{8}) \\ \ln (\cos (- \frac{蟺}{8}) + \sin (- \frac{蟺}{8})) \end{array}] + 3 \\ \ln (- \frac{蟺}{8}) \end{array}\} \end{array}]$

$m = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

Thus, the mass of semicircular lamina is

m = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))

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91影视

Complete Example 2 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} \frac{k}{3} (8 \cos^{3} 胃 - \sec^{3} 胃) d 胃 = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

Short Answer

Step by step solution

Given information

Calculation

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91影视

Complete Example 2 by showing that鈭�-蟺/4蟺/4k38cos3胃-sec3胃d胃=k9(172+3ln(2-1))

Short Answer

Step by step solution

Given information

Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Mechanics Maths

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Complete Example 2 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} \frac{k}{3} (8 \cos^{3} 胃 - \sec^{3} 胃) d 胃 = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$