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.
$\int \int_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{π}{4} \leq x \leq \frac{π}{2} a n d \frac{π}{2} \leq y \leq π\}$

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,

$\int \int_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{π}{4} \leq x \leq \frac{π}{2} a n d \frac{π}{2} \leq y \leq π\}$

Step 2. Explanation.

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

$\int \int_{R} \cos (x y) d A = \int_{\frac{π}{4}}^{\frac{π}{2}} \int_{\frac{π}{2}}^{π} \cos (x y) d y d x = \int_{\frac{π}{4}}^{\frac{π}{2}} (\int_{\frac{π}{2}}^{π} \cos (x y) d y) d x = \int_{\frac{π}{4}}^{\frac{π}{2}} {[\frac{1}{x} \sin (x y)]}_{y = \frac{π}{2}}^{y = π} d x = \int_{\frac{π}{4}}^{\frac{π}{2}} [\frac{1}{x} \sin (πx) - \frac{1}{x} \sin (\frac{π}{2} x)] d x$

Now the resultant function $\frac{1}{x} \sin (πx) - \frac{1}{x} \sin (\frac{π}{2} x)$ 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.
$\int \int_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{π}{4} \leq x \leq \frac{π}{2} a n d \frac{π}{2} \leq y \leq π\}$

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.∫∫RcosxydAwhereR=x,yIπ4≤x≤π2andπ2≤y≤π

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Probability and Statistics

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

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.
$\int \int_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{π}{4} \leq x \leq \frac{π}{2} a n d \frac{π}{2} \leq y \leq π\}$