Q. 25 Each of the integrals or integra... [FREE SOLUTION]

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Chapter 13: Q. 25 (page 1024)

Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃$

Short Answer

Expert verified

The integral's value is $2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃 = \frac{蟺}{4}$

Step by step solution

given information

Let consider the given integral is $2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� r d r d 胃$

Finding establish the expression

The goal of this issue is to sketch the region and assess the expression using polar coordinates $2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� r d r d 胃$

Using, $r = 0, r = \sin 鈦� 胃 and 胃 = 0, 胃 = 蟺 / 2$

$胃$	$r = \sin 3 胃$
0	0
$蟺 / 6$	$1.0$
$蟺 / 4$	$0.7071$
$蟺 / 3$	$0$
$蟺 / 2$	$- 1.0$

Calculations

$\begin{aligned} 2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃 = 2 {鈭�}_{0}^{蟺 / 2} 鈥� {[\frac{r^{2}}{2}]}_{0}^{\sin 鈦� 3 胃} 诲胃 \\ = 2 {鈭�}_{0}^{蟺 / 2} 鈥� [\frac{\sin^{2} 鈦� 3 胃鈭� 0}{2}] 胃 \\ = 2 {鈭�}_{0}^{蟺 / 2} 鈥� [\frac{(1 鈭� \cos 鈦� 6 胃)}{4}] 诲胃 \end{aligned}$

Integrate in relation to $胃$ ,

$2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃 = 2 {[\frac{{\frac{(胃鈭� (\sin 鈦� 6 胃)}{6}}}{4}]}_{0}^{蟺 / 2} [鈭� \cos 鈦� xdx = \sin 鈦� x]$

Pointing the limits,

$2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃 = 2 [\frac{{\frac{蟺 / 2 鈭� (\sin 鈦� 3 蟺)}{6} 鈭� 0}}{4}] = \frac{蟺}{4}$

As a result, the integral value is $2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃 = \frac{蟺}{4}$

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91影视

Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃$

Short Answer

Step by step solution

given information

Finding establish the expression

Calculations

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91影视

Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.2鈭�0蟺/2鈥�鈭�0sin鈦�3胃鈥�谤诲谤诲胃

Short Answer

Step by step solution

given information

Finding establish the expression

Calculations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Statistics

Calculus

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.
$2 {鈭�}_{0}^{蟺 / 2} 鈥� {鈭�}_{0}^{\sin 鈦� 3 胃} 鈥� 谤诲谤诲胃$