Q. 23 Complete Example 3 by showing th... [FREE SOLUTION]

Chapter 13: Q. 23 (page 1038)

Complete Example 3 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} k r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1)) .$

Short Answer

Expert verified

The first moment of the mass about y-axis is

$M_{y} = k {鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1))$

Step by step solution

Given information

The expression is

{鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} k r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \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 $r = \sec 胃$

First moment of the mass about $y$ - axis is

$M_{y} = {鈭�}_{惟} x 蚁 (x, y) d A M_{y} = {鈭�}_{惟} k r^{3} d r d 胃 M_{y} = k {鈭�}_{- 蟺 / 4}^{n / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} r^{3} \cos 胃 d r d 胃$

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

$M_{y} = k {鈭�}_{- 蟺 / 4}^{蟺 / 4} {[\frac{r^{4}}{4}]}_{\sec 胃}^{2 \cos 胃} \cos 胃 d 胃$

Substitute the limits

$M_{y} = \frac{k}{4} {鈭�}_{- 蟺 / 4}^{蟺 / 4} [16 \cos^{4} 胃 \sec^{4} 胃] d 胃$

Integrate with respect to $胃 .$

M_{y} = \frac{k}{4} {[\frac{16}{32} {12 胃 + 8 \sin 2 胃 + \sin 4 胃} - \frac{1}{3} \{(\cos 2 胃 + 2) \tan 胃 \sec^{2} 胃\}]}_{- 蟺 / 4}^{蟺 / 4} M_{y} = \frac{k}{4} [\frac{1}{2} {12 (蟺 / 4) + 8 \sin (蟺 / 2) + \sin (蟺)} - \frac{1}{3} \{(\cos (蟺 / 2) + 2) \tan (蟺 / 4) \sec^{2} (蟺 / 4)\}] - \frac{k}{4} [\frac{1}{2} {12 (- 蟺 / 4) + 8 \sin (- 蟺 / 2) + \sin (- 蟺)} - \frac{1}{3} \{(\cos (- 蟺 / 2) + 2) \tan (- 蟺 / 4) \sec^{2} (- 蟺 / 4)\}] M_{y} = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1))

Thus, the first moment of the mass about $y$ - axis is

$M_{y} = k {鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 eos 胃} r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1))$

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91影视

Complete Example 3 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} k r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1)) .$

Short Answer

Step by step solution

Given information

Calculation

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91影视

Complete Example 3 by showing that鈭�-蟺/4蟺/4鈭�sec胃2cos胃kr3cos胃drd胃=k60(1572+15ln(2-1)).

Short Answer

Step by step solution

Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Probability and Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Complete Example 3 by showing that
${鈭�}_{- 蟺 / 4}^{蟺 / 4} {鈭�}_{\sec 胃}^{2 \cos 胃} k r^{3} \cos 胃 d r d 胃 = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1)) .$