Chapter 13: Q. 22 (page 1038)

Complete Example 2 by showing that
$\int_{- π / 4}^{π / 4} \frac{k}{3} (8 \cos^{3} θ - \sec^{3} θ) d θ = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

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

\int_{- π / 4}^{π / 4} \frac{k}{3} (8 \cos^{3} θ - \sec^{3} θ) d θ = \frac{k}{9} (17 \sqrt{2} + 3 \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 $r = \sec θ$

Mass of the plate

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

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

$m = k \int_{- π / 4}^{π / 4} {[\frac{r^{3}}{3}]}_{\sec θ}^{2 \cos θ} d θ m = k \int_{- n / 4}^{π / 4} [\frac{8 \cos^{3} θ - \sec^{3} θ}{3}] d θ$

Substitute the limits

$m = \frac{k}{3} \int_{- π / 4}^{π / 4} [8 \cos^{3} θ - \sec^{3} θ] d θ$

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

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

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

Thus, the mass of semicircular lamina is

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

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Complete Example 2 by showing that
$\int_{- π / 4}^{π / 4} \frac{k}{3} (8 \cos^{3} θ - \sec^{3} θ) d θ = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$

Short Answer

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Calculation

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Complete Example 2 by showing that∫-π/4π/4k38cos3θ-sec3θdθ=k9(172+3ln(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

Probability and Statistics

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Complete Example 2 by showing that
$\int_{- π / 4}^{π / 4} \frac{k}{3} (8 \cos^{3} θ - \sec^{3} θ) d θ = \frac{k}{9} (17 \sqrt{2} + 3 \ln (\sqrt{2} - 1))$