Q. 22 Each of the integrals or integra... [FREE SOLUTION]

91影视

Chapter 13: Q. 22 (page 1027)

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

Short Answer

Expert verified

The value of integer is $2 {鈭�}_{0}^{\frac{蟺}{2}} 鈥� {鈭�}_{0}^{\sin 鈦� 胃} 鈥� 谤诲谤诲胃 = \frac{蟺}{4}$

Step by step solution

Given information

The function is $2 {鈭�}_{0}^{\frac{蟺}{2}} 鈥� {鈭�}_{0}^{\sin 鈦� 胃} 鈥� 谤诲谤诲胃$

Calculation of integrals

The goal of this issue is to sketch the region and assess the expression using polar coordinates $2 {鈭�}_{0}^{蟺} 鈥� {鈭�}_{0}^{1 + \cos 鈦� 胃} 鈥� 谤诲谤诲胃$

using the values , $r = 0, r = 1 + \cos 鈦� 胃 and 胃 = 0, 胃 = 蟺$

Use the table above to draw the region $2 {鈭�}_{0}^{蟺} 鈥� {鈭�}_{0}^{1 + \cos 鈦� 胃} 鈥� 谤诲谤诲胃 = 2 {鈭�}_{0}^{蟺} 鈥� {[\frac{r^{2}}{2}]}_{0}^{1 + \cos 鈦� 胃} 诲胃 = 2 {鈭�}_{0}^{蟺} 鈥� [\frac{(1 + \cos 鈦� 胃)^{2} 鈭� 0}{2}] 诲胃 = 2 {鈭�}_{0}^{蟺} 鈥� [\frac{(1 + 2 \cos 鈦� 胃 + \cos^{2} 鈦� 胃)}{2}] 诲胃 [(a + b)^{2} = a^{2} + 2 ab + b^{2}] = 2 {鈭�}_{0}^{蟺} 鈥� [\frac{\{1 + 2 \cos 鈦� 胃 + \frac{(1 + \cos 鈦� 2 胃)}{2}\}}{2}] 诲胃 = 2 {鈭�}_{0}^{x} 鈥� [\frac{3 + 4 \cos 鈦� 胃 + \cos 鈦� 2 胃}{4}] 诲胃$

We can integrating in terms of $胃$

$2 {鈭�}_{0}^{蟺} 鈥� {鈭�}_{0}^{1 + \cos 鈦� 胃} 鈥� 谤诲谤诲胃 = 2 {[\frac{3 胃 + 4 \sin 鈦� 胃 + \frac{(\sin 鈦� 2 胃)}{2}}{4}]}_{0}^{蟺}$

$[鈭� \sin 鈦� xdx = 鈭� \cos 鈦� x, 鈭� \cos 鈦� xdx = \sin 鈦� x]$

Substituting the limits of derivatives,

$2 {鈭�}_{0}^{x} 鈥� {鈭�}_{0}^{1 \cos 鈦� 胃} 鈥� 谤诲谤诲胃 = 2 [\frac{3 蟺 + 4 \sin 鈦� 蟺 + \frac{(\sin 鈦� 2 蟺)}{2} 鈭� 0}{4}]$

As a result, the integral value is ${鈭�}_{0}^{2 蟺} 鈥� {鈭�}_{0}^{1 + \sin 鈦� 胃} 鈥� 谤诲谤诲胃 = \frac{3 蟺}{2}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Given information

Calculation of integrals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Each of the integrals or integral expressions in Exercises 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�胃鈥�谤诲谤诲胃

Short Answer

Step by step solution

Given information

Calculation of integrals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

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