Chapter 13: Q. 42 (page 1004)

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | - 3 \leq x \leq 2 a n d - 2 \leq y \leq 2}$

Short Answer

Expert verified

The value is $- 4 \sin (2) \cos (2) + 2 \sin^{2} 2 - 6 \sin (2) \cos (3) + 2 \sin (2) \sin (3)$

Step by step solution

Step 1. Given Information :

Given double integrals :

$\int \int_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | - 3 \leq x \leq 2 a n d - 2 \leq y \leq 2}$

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

Step 2. Solution:

$U \sin g F u b i n i' s T h e o r e m \int \int_{R} x \sin x \cos y d A = \int_{- 2}^{2} \int_{- 3}^{2} x \sin x \cos y d x d y E v a l u a t i o n p r o c e d u r e f o r t h e i t e r a t e d i n t e g r a l w e g e t = \int_{- 2}^{2} (\int_{- 3}^{2} x \sin x \cos y d x) d y$

$B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = \int_{- 2}^{2} (\cos y ({[- x \cos x]}_{- 3}^{2} + {[\sin x]}_{- 3}^{2})) d y E v a l u a t i o n o f t h e i n n e r a n t i d e r i v a t i v e w e g e t = \int_{- 2}^{2} \cos y [(- 2 \cos 2 - 3 \cos 3) + (\sin 2 + \sin 3)] d y = \int_{- 2}^{2} \cos y [(- 2 \cos 2 + \sin 2 - 3 \cos 3 + \sin 3)] d y$

$B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = [(- 2 \cos 2 + \sin 2 - 3 \cos 3 + \sin 3)] {[\sin y]}_{- 2}^{2} E v a l u a t i o n o f t h e o u t e r a n t i d e r i v a t i v e w e g e t = [(- 2 \cos 2 + \sin 2 - 3 \cos 3 + \sin 3)] [\sin 2 + \sin 2] = [(- 2 \cos 2 + \sin 2 - 3 \cos 3 + \sin 3)] [2 \sin 2] = [- 4 \sin (2) \cos (2) + 2 \sin^{2} 2 - 6 \sin (2) \cos (3) + 2 \sin (2) \sin (3)]$

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Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | - 3 \leq x \leq 2 a n d - 2 \leq y \leq 2}$

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.∫∫RxsinxcosydA,whereR={(x,y)|−3≤x≤2and−2≤y≤2}

Short Answer

Step by step solution

Step 1. Given Information :

Step 2. Solution:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.
$\int \int_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | - 3 \leq x \leq 2 a n d - 2 \leq y \leq 2}$