Q. 45 Evaluate each of the double inte... [FREE SOLUTION]

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Chapter 13: Q. 45 (page 1004)

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 鈮� x 鈮� 4 a n d 0 鈮� y 鈮� 3}$

Short Answer

Expert verified

The value is $- 8 \ln (2) + \frac{9}{2} + \frac{7}{2} \ln (7)$

Step by step solution

Step 1. Given Information:

Given double integrals :

$鈭� {鈭�}_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 鈮� x 鈮� 4 a n d 0 鈮� y 鈮� 3}$

We want to evaluate each of the double integrals as iterated integrals.

Step 2. Solution:

Using Fubini's Theorem

鈭� {鈭�}_{R} \frac{x}{x + y} d A = {鈭�}_{1}^{4} {鈭�}_{0}^{3} \frac{x}{x + y} d y d x E v a l u a t i o n p r o c e d u r e f o r t h e i t e r a t e d i n t e g r a l w e g e t = {鈭�}_{1}^{4} ({鈭�}_{0}^{3} \frac{x}{x + y} d y) d x = {鈭�}_{1}^{4} x {鈭�}_{0}^{3} \frac{1}{x + y} d y d x

First we solve:

${鈭�}_{0}^{3} \frac{1}{x + y} d y = {[\ln (x + t)]}_{0}^{3} = \ln (x + 3) - \ln x S o i n t e g r a l b e c o m e s : = {鈭�}_{1}^{4} x [\ln (x + 3) - \ln x] d x = {鈭�}_{1}^{4} x \ln (x + 3) d x - {鈭�}_{1}^{4} x \ln x d x = I_{1} - I_{2} S o l v e I_{1} w e h a v e P u t x + 3 = t s o w e g e t d x = d t w h e n x = 4 t h e n t = 7 w h e n x = 1 t h e n t = 4 I_{1} b e c o m e s {鈭�}_{4}^{7} (t - 3) \ln (t) d t A p p l y i n t e g r a t i o n B y p a r t s : = {[\ln (t) (\frac{t^{2}}{2} - 3 t) - 鈭� \frac{1}{2} (t - 6) d t]}_{4}^{7} = {[\ln (t) (\frac{t^{2}}{2} - 3 t) - \frac{1}{2} (\frac{t^{2}}{2} - 6 t)]}_{4}^{7} = 8 \ln (2) + \frac{3}{4} + \frac{7}{2} l n (7) S o l v e I_{2} w e h a v e {鈭�}_{1}^{4} x \ln x d x A p p l y i n t e g r a t i o n B y p a r t s : = {[\ln (x) \frac{x^{2}}{2} - 鈭� \frac{x}{2} d x]}_{1}^{4} = {[\ln (x) \frac{x^{2}}{2} - \frac{x^{2}}{4}]}_{1}^{4} = 16 \ln (2) - \frac{15}{4} S o w e h a v e I_{1} - I_{2} = 8 \ln (2) + \frac{3}{4} + \frac{7}{2} l n (7) - 16 \ln (2) + \frac{15}{4} = - 8 \ln (2) + \frac{9}{2} + \frac{7}{2} \ln (7)$

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91影视

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 鈮� x 鈮� 4 a n d 0 鈮� y 鈮� 3}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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91影视

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.鈭�鈭�Rxx+ydA,whereR={(x,y)|1鈮�x鈮�4and0鈮�y鈮�3}

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 鈮� x 鈮� 4 a n d 0 鈮� y 鈮� 3}$