Q. 43 Evaluate each of the double inte... [FREE SOLUTION]

91影视

Chapter 13: Q. 43 (page 1004)

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 鈮� x 鈮� 1 a n d 0 鈮� y 鈮� l n 5}$

Short Answer

Expert verified

The value is $\frac{4}{\ln 5} - 1$

Step by step solution

Step 1. Given Information:

Given double integrals :

$鈭� {鈭�}_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 鈮� x 鈮� 1 a n d 0 鈮� y 鈮� l n 5}$

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

Step 2. Solution:

$U \sin g F u b i n i' s T h e o r e m 鈭� {鈭�}_{R} x e^{x y} d A = {鈭�}_{0}^{1} {鈭�}_{0}^{l n 5} x e^{x y} d y d x E v a l u a t i o n p r o c e d u r e f o r t h e i t e r a t e d i n t e g r a l w e g e t = {鈭�}_{0}^{1} ({鈭�}_{0}^{\ln 5} x e^{x y} d y) d x = {鈭�}_{0}^{1} x ({鈭�}_{0}^{\ln 5} e^{x y} d y) d x$

$F i r s t w e s o l v e {鈭�}_{0}^{\ln 5} e^{x y} d y P u t x y = t s o w e h a v e d y = \frac{d t}{x} W h e n y = \ln (5) t h e n t = x \ln (5) W h e n y = 0 t h e n t = 0 S o i n t e g r a l b e c o m e s : = {鈭�}_{0}^{x \ln 5} \frac{e^{t}}{x} d t = \frac{1}{x} {鈭�}_{0}^{x \ln 5} e^{t} d t B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = \frac{1}{x} {[e^{t}]}_{0}^{x \ln 5} E v a l u a t i o n o f t h e i n n e r a n t i d e r i v a t i v e w e g e t = \frac{1}{x} [e^{\ln 5^{x}} - 1] S o w e h a v e = {鈭�}_{0}^{1} x \frac{1}{x} [5^{x} - 1] d x = {鈭�}_{0}^{1} [5^{x} - 1] d x$

$B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = {鈭�}_{0}^{1} 5^{x} d x - {鈭�}_{0}^{1} d x = {鈭�}_{0}^{1} e^{\ln 5^{x}} d x - {鈭�}_{0}^{1} d x = {鈭�}_{0}^{1} e^{x \ln 5} d x - {鈭�}_{0}^{1} d x = {[\frac{e^{x \ln 5}}{\ln 5}]}_{0}^{1} - {[x]}_{0}^{1} = \frac{e^{\ln 5} - 1}{\ln 5} - 1 = \frac{4}{\ln 5} - 1$

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

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 鈮� x 鈮� 1 a n d 0 鈮� y 鈮� l n 5}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Logic and Functions

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.鈭�鈭�RxexydA,where{R=(x,y)|0鈮�x鈮�1and0鈮�y鈮�ln5}

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Logic and Functions

Decision Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.
$鈭� {鈭�}_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 鈮� x 鈮� 1 a n d 0 鈮� y 鈮� l n 5}$