Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 14 - Multiple Integration - 14.2 Exercises - Page 984: 56

Answer

$$0$$

Work Step by Step

$$\eqalign{ & f\left( {x,y} \right) = \sin \left( {x + y} \right) \cr & {\text{From the image of the rectangle shown below we obtain}} \cr & {\text{the region }}R \cr & R = \left\{ {\left. {\left( {x,y} \right)} \right|0 \leqslant x \leqslant \pi ,{\text{ }}x \leqslant y \leqslant \pi } \right\} \cr & {\text{The area }}A{\text{ of the region is:}} \cr & A = \left( \pi \right)\left( \pi \right) \cr & A = {\pi ^2} \cr & {\text{The average Value of a Function Over a Region }}R{\text{ is}} \cr & {\text{Average value}} = \frac{1}{A}\iint\limits_R {f\left( {x,y} \right)}dA,{\text{ then}} \cr & {\text{Average value}} = \frac{1}{{{\pi ^2}}}\int_0^\pi {\int_0^\pi {\sin \left( {x + y} \right)} dydx} \cr & = \frac{1}{{{\pi ^2}}}\int_0^\pi {\left[ {\int_0^\pi {\sin \left( {x + y} \right)} dy} \right]dx} \cr & {\text{Integrate with respect to }}y \cr & = - \frac{1}{{{\pi ^2}}}\int_0^\pi {\left[ {\cos \left( {x + y} \right)} \right]_0^\pi dx} \cr & = - \frac{1}{{{\pi ^2}}}\int_0^\pi {\left[ {\cos \left( {x + \pi } \right) - \cos \left( {x + 0} \right)} \right]dx} \cr & = - \frac{1}{{{\pi ^2}}}\int_0^\pi {\left[ {\cos \left( {x + \pi } \right) - \cos x} \right]dx} \cr & {\text{Integrate}} \cr & = - \frac{1}{{{\pi ^2}}}\left[ { - \sin \left( {x + \pi } \right) + \sin x} \right]_0^\pi \cr & = - \frac{1}{{{\pi ^2}}}\left[ { - \sin \left( {\pi + \pi } \right) + \sin \pi } \right] + \frac{1}{{{\pi ^2}}}\left[ { - \sin \left( {0 + \pi } \right) + \sin 0} \right] \cr & = - \frac{1}{{{\pi ^2}}}\left[ { - \sin \left( {2\pi } \right) + \sin \pi } \right] + \frac{1}{{{\pi ^2}}}\left[ { - \sin \left( \pi \right) + \sin 0} \right] \cr & = 0 \cr} $$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.