Chapter 13: Q 53 (page 1005)

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
$\int \int_{R} x^{2} e^{x y} d A,$
where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$ .

Short Answer

Expert verified

The value of double integral is :-

$\int \int_{R} x^{2} e^{x y} d A = \frac{1}{2}$

where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$

Step by step solution

Step 1. Given Information

We have given the following double integral :-

$\int \int_{R} x^{2} e^{x y} d A,$

where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$

We have to evaluate this double integral.

Step 2. Use iterated integrals

The given double integral is :-

$\int \int_{R} x^{2} e^{x y} d A,$

where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$

In order to solve this double integral we will firstly integrated with $y$ .

Then by using Fubini's Theorem, we can writ this double integral as following :-

role="math" localid="1650586863367" $\int \int_{R} x^{2} e^{x y} d A = \int_{0}^{1} {\int_{0}^{1}}_{R} x^{2} e^{x y} d y d x$

Then by using iterated integrals, we have :-

$\int_{0}^{1} {\int_{0}^{1}}_{R} x^{2} e^{x y} d y d x = \int_{0}^{1} ({\int_{0}^{1}}_{R} x^{2} e^{x y} d y) d x$

Now we can solve this integral as following :-

$\int_{0}^{1} ({\int_{0}^{1}}_{R} x^{2} e^{x y} d y) d x = \int_{0}^{1} x^{2} {[\frac{e^{x y}}{x}]}^{1}_{0} d x = \int_{0}^{1} \frac{x^{2}}{x} [e^{x} - e^{0}] d x = \int_{0}^{1} x [e^{x} - 1] d x = \int_{0}^{1} (x e^{x} - x) d x$

Now use product rule of integration $\int f (x) g (x) d x = f (x) \int g (x) d x - \int [\frac{d}{d x} f (x) (\int g (x)) d x] d x$

$= {[(x e^{x} - e^{x}) - \frac{x^{2}}{2}]}^{1}_{0} = [(e - e - \frac{1}{2}) - (0 - e^{0} - 0)] = - \frac{1}{2} + 1 = \frac{1}{2}$

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Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
$\int \int_{R} x^{2} e^{x y} d A,$
where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use iterated integrals

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Evaluate each of the double integrals in Exercises 37-54as iterated integrals.∫∫Rx2exydA,whereR=x,y|0≤x≤1and0≤y≤1.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use iterated integrals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.
$\int \int_{R} x^{2} e^{x y} d A,$
where $R = \{(x, y) | 0 \leq x \leq 1 a n d 0 \leq y \leq 1\}$ .