Chapter 13: Q. 23 (page 1038)

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

Short Answer

Expert verified

The first moment of the mass about y-axis is

$M_{y} = k \int_{- π / 4}^{π / 4} \int_{\sec θ}^{2 \cos θ} r^{3} \cos θ d r d θ = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1))$

Step by step solution

Given information

The expression is

\int_{- π / 4}^{π / 4} \int_{\sec θ}^{2 \cos θ} k r^{3} \cos θ d r d θ = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1)) .

Calculation

Density $ρ (r) = k r$

Equation of circle

$r = 2 \cos θ$

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

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

$M_{y} = \iint_{Ω} x ρ (x, y) d A M_{y} = \iint_{Ω} k r^{3} d r d θ M_{y} = k \int_{- π / 4}^{n / 4} \int_{\sec θ}^{2 \cos θ} r^{3} \cos θ d r d θ$

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

$M_{y} = k \int_{- π / 4}^{π / 4} {[\frac{r^{4}}{4}]}_{\sec θ}^{2 \cos θ} \cos θ d θ$

Substitute the limits

$M_{y} = \frac{k}{4} \int_{- π / 4}^{π / 4} [16 \cos^{4} θ \sec^{4} θ] d θ$

Integrate with respect to $θ .$

M_{y} = \frac{k}{4} {[\frac{16}{32} {12 θ + 8 \sin 2 θ + \sin 4 θ} - \frac{1}{3} \{(\cos 2 θ + 2) \tan θ \sec^{2} θ\}]}_{- π / 4}^{π / 4} M_{y} = \frac{k}{4} [\frac{1}{2} {12 (π / 4) + 8 \sin (π / 2) + \sin (π)} - \frac{1}{3} \{(\cos (π / 2) + 2) \tan (π / 4) \sec^{2} (π / 4)\}] - \frac{k}{4} [\frac{1}{2} {12 (- π / 4) + 8 \sin (- π / 2) + \sin (- π)} - \frac{1}{3} \{(\cos (- π / 2) + 2) \tan (- π / 4) \sec^{2} (- π / 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

$M_{y} = k \int_{- π / 4}^{π / 4} \int_{\sec θ}^{2 eos θ} r^{3} \cos θ d r d θ = \frac{k}{60} (157 \sqrt{2} + 15 \ln (\sqrt{2} - 1))$

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

Short Answer

Step by step solution

Given information

Calculation

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Complete Example 3 by showing that∫-π/4π/4∫secθ2cosθkr3cosθdrdθ=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

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

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