Q. 19 Explain why it would be difficul... [FREE SOLUTION]

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Chapter 13: Q. 19 (page 1004)

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Short Answer

Expert verified

The double integral is difficult to evaluate using iterated integral.

Step by step solution

Step 1. Given Information.

The iterated integral is,

$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Step 2. Explanation.

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

$鈭� {鈭�}_{R} \cos (x y) d A = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} {鈭�}_{\frac{蟺}{2}}^{蟺} \cos (x y) d y d x = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} ({鈭�}_{\frac{蟺}{2}}^{蟺} \cos (x y) d y) d x = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} {[\frac{1}{x} \sin (x y)]}_{y = \frac{蟺}{2}}^{y = 蟺} d x = {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} [\frac{1}{x} \sin (蟺虫) - \frac{1}{x} \sin (\frac{蟺}{2} x)] d x$

Now the resultant function $\frac{1}{x} \sin (蟺虫) - \frac{1}{x} \sin (\frac{蟺}{2} x)$ does not have a simple anti-derivative to evaluate the next step which is integrating with respect to variable $x$ . The double integral is difficult to evaluate using iterated integral.

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Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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91影视

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.鈭�鈭�RcosxydAwhereR=x,yI蟺4鈮�x鈮�蟺2and蟺2鈮�y鈮�蟺

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Applied Mathematics

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.
$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$