Chapter 13: Q. 43 (page 1004)

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq l n 5}$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

Given double integrals :

$\int \int_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq l n 5}$

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

Step 2. Solution:

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

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

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

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Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq l n 5}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.∫∫RxexydA,where{R=(x,y)|0≤x≤1and0≤y≤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

Geometry

Pure Maths

Logic and Functions

Theoretical and Mathematical Physics

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Statistics

Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq l n 5}$