Chapter 13: Q. 18 (page 1004)

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
$\int \int_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 \leq x \leq 2 a n d 1 \leq y \leq 3\}$

Short Answer

Expert verified

The given double integral is difficult to evaluate using iterated integrals.

Step by step solution

Step 1. Given Information.

The iterated integral is,

$\int \int_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 \leq x \leq 2 a n d 1 \leq y \leq 3\}$

Step 2. Explanation.

Evaluating the first integral with respect to $y$ and treating $x$ as constant.

$\int \int_{R} e^{x y} d A = \int_{1}^{2} \int_{1}^{3} e^{x y} d y d x = \int_{1}^{2} (\int_{1}^{3} e^{x y} d y) d x = \int_{1}^{2} {[\frac{1}{x} e^{x y}]}_{y = 1}^{y = 3} d x = \int_{1}^{2} [\frac{1}{x} e^{3 x} - \frac{1}{x} e^{x}] d x$

Now the resultant function $\frac{1}{x} e^{3 x} - \frac{1}{x} e^{x}$ does not have a simple anti-derivative to evaluate the next step which is integrating with respect to variable $x$ . The given double integral is difficult to evaluate using iterated integrals.

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Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
$\int \int_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 \leq x \leq 2 a n d 1 \leq y \leq 3\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.∫∫RexydAwhereR=x,yI1≤x≤2and1≤y≤3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Most popular questions from this chapter

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Pure Maths

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Study anywhere. Anytime. Across all devices.

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
$\int \int_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 \leq x \leq 2 a n d 1 \leq y \leq 3\}$