Chapter 13: Q. 45 (page 1027)

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3} .$ Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ$

Short Answer

Expert verified

The value of integral is

$\int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ = \frac{3}{4} (π + 1)$

Step by step solution

Given information

The expression is

I = \int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ

Calculation

Here, $r = 0, r = (\sqrt{2} / 2) + \sin θ a n d θ = - π / 4, θ = 5 π / 4$

Integrate with respect to $r$ first,

$I = \int_{- π / 4}^{5 π / 4} {[\frac{r^{2}}{2}]}_{0}^{(\sqrt{2} / 2) + \sin θ} d θ [\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Put the limits

$I = \int_{- π / 4}^{5 π / 4} [\frac{{(\sqrt{2} / 2) + \sin θ}^{2} - 0}{2}] d θ I = \int_{- π / 4}^{5 π / 4} [\frac{\{\frac{1}{2} + \sqrt{2} \sin θ + \sin^{2} θ\}}{2}] d θ I = \int_{- π / 4}^{5 π / 4} [\frac{\{\frac{1}{2} + \sqrt{2} \sin θ + \frac{1}{2} (1 - \cos 2 θ)\}}{2}] d θ$

Now integrate with respect to $θ$

$I = {[\frac{\{\frac{θ}{2} - \sqrt{2} \cos θ + \frac{1}{2} (θ - \frac{1}{2} \sin 2 θ)\}}{2}]}_{- π / 4}^{3 π / 4}$

$I = [\frac{{\{θ - \sqrt{2} \cos θ - \frac{1}{4} \sin 2 θ\}]}_{- π / 4}^{5 π / 4}}{2}] I = [\frac{\{\frac{5 π}{4} - \sqrt{2} \cos (\frac{5 π}{4}) - \frac{1}{4} \sin (\frac{5 π}{2})\} - \{\frac{- π}{4} - \sqrt{2} \cos (\frac{- π}{4}) - \frac{1}{4} \sin (\frac{- π}{2})\}}{2}]$

$I = [\frac{\{\frac{5 π}{4} - \sqrt{2} (- \frac{1}{\sqrt{2}}) - \frac{1}{4}\} - \{\frac{- π}{4} - \sqrt{2} (\frac{1}{\sqrt{2}}) + \frac{1}{4}\}}{2}] I = [\frac{\{\frac{5 π}{4} + 1 - \frac{1}{4}\} - \{\frac{- π}{4} - 1 + \frac{1}{4}\}}{2}] I = \frac{3}{4} (π + 1)$

Thus, the value of integral is

localid="1651500227368" $\int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ = \frac{3}{4} (π + 1)$

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Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3} .$ Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ$

Short Answer

Step by step solution

Given information

Calculation

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Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in ℝ3.Use polar coordinates to describe the solid, and evaluate the expressions.∫-π/45π/4∫0(2/2)+sinθrdrdθ

Short Answer

Step by step solution

Given information

Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

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Statistics

Study anywhere. Anytime. Across all devices.

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3} .$ Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{- π / 4}^{5 π / 4} \int_{0}^{(\sqrt{2} / 2) + \sin θ} r d r d θ$