Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 16 - Multiple Integration - 16.2 Double Integrals over More General Regions - Exercises - Page 859: 41


$6-6 \ln (2) $

Work Step by Step

The iterated integral can be calculated as: $\iint_{D} f(x,y) d A=\int_2^4 \int_{y-1}^{7-y} \dfrac{x}{y^2}dx dy\\=\int_2^4 (\dfrac{x^2}{2y^2})_{y-1}^{7-y} dx \\=\int_2^4 [\dfrac{(7-y)^2}{2y^2}-\dfrac{(y-1)^2}{2y^2}] \ dy \\=\int_2^4 (\dfrac{24}{y^2}-\dfrac{6}{y}] \ dy \\=[\dfrac{-24}{y}-6 \ln y ]_2^4\\=\dfrac{-24}{4}-6 \ln 4 +\dfrac{24}{2}+6 \ln 2 \\=6-6 \ln (2) $
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