Chapter 13: Q. 44 (page 1004)

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

Short Answer

Expert verified

The value is $π$

Step by step solution

Step 1. Given Information:

Given double integrals :

$\int \int_{R} x^{2} \cos (x y) d A, w h e r e R = {(x, y) | 0 \leq x \leq π a n d 0 \leq y \leq 1}$

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

Step 2. Solution:

$U \sin g F u b i n i' s T h e o r e m \int \int_{R} x^{2} \cos (x y) d A = \int_{0}^{π} \int_{0}^{1} x^{2} \cos (x y) d y d x E v a l u a t i o n p r o c e d u r e f o r t h e i t e r a t e d i n t e g r a l w e g e t = \int_{0}^{π} (\int_{0}^{1} x^{2} \cos (x y) d y) d x = \int_{0}^{π} (x^{2} \int_{0}^{1} \cos (x y) d y) d x$

$F i r s t w e s o l v e \int_{0}^{1} \cos (x y) d y P u t x y = t s o w e h a v e d y = \frac{d t}{x} W h e n y = 1 t h e n t = x W h e n y = 0 t h e n t = 0 S o i n t e g r a l b e c o m e s : = \int_{0}^{x} \frac{\cos t}{x} d t B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = \frac{1}{x} {[\sin t]}_{0}^{x} E v a l u a t i o n o f t h e i n n e r a n t i d e r i v a t i v e w e g e t = \frac{\sin x}{x} S o w e h a v e = \int_{0}^{π} x^{2} \frac{\sin x}{x} d x = \int_{0}^{π} x \sin x d x$

$B y F u n d a m e n t a l T h e o r e m o f C a l c u l u s w e h a v e = {[x (- \cos x)]}_{0}^{π} - \int_{0}^{π} - \cos x d x = {[x (- \cos x)]}_{0}^{π} + {[\sin x]}_{0}^{π} = (- π cosπ - 0) + (sinπ - 0) = π$

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

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.∫∫Rx2cos(xy)dA,whereR={(x,y)|0≤x≤πand0≤y≤1}

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

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Decision Maths

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Study anywhere. Anytime. Across all devices.

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