Q. 45 Each of the integrals or integra... [FREE SOLUTION]

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Chapter 13: Q. 45 (page 1027)

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

Short Answer

Expert verified

The value of integral is

${鈭�}_{- 蟺 / 4}^{5 蟺 / 4} {鈭�}_{0}^{(\sqrt{2} / 2) + \sin 胃} r d r d 胃 = \frac{3}{4} (蟺 + 1)$

Step by step solution

Given information

The expression is

I = {鈭�}_{- 蟺 / 4}^{5 蟺 / 4} {鈭�}_{0}^{(\sqrt{2} / 2) + \sin 胃} r d r d 胃

Calculation

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

Integrate with respect to $r$ first,

$I = {鈭�}_{- 蟺 / 4}^{5 蟺 / 4} {[\frac{r^{2}}{2}]}_{0}^{(\sqrt{2} / 2) + \sin 胃} d 胃 [鈭� x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Put the limits

$I = {鈭�}_{- 蟺 / 4}^{5 蟺 / 4} [\frac{{(\sqrt{2} / 2) + \sin 胃}^{2} - 0}{2}] d 胃 I = {鈭�}_{- 蟺 / 4}^{5 蟺 / 4} [\frac{\{\frac{1}{2} + \sqrt{2} \sin 胃 + \sin^{2} 胃\}}{2}] d 胃 I = {鈭�}_{- 蟺 / 4}^{5 蟺 / 4} [\frac{\{\frac{1}{2} + \sqrt{2} \sin 胃 + \frac{1}{2} (1 - \cos 2 胃)\}}{2}] d 胃$

Now integrate with respect to $胃$

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

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

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

Thus, the value of integral is

localid="1651500227368" ${鈭�}_{- 蟺 / 4}^{5 蟺 / 4} {鈭�}_{0}^{(\sqrt{2} / 2) + \sin 胃} r d r d 胃 = \frac{3}{4} (蟺 + 1)$

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91影视

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

Short Answer

Step by step solution

Given information

Calculation

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91影视

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in 鈩�3.Use polar coordinates to describe the solid, and evaluate the expressions.鈭�-蟺/45蟺/4鈭�0(2/2)+sin胃rdrd胃

Short Answer

Step by step solution

Given information

Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Geometry

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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