Calculus (3rd Edition)

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

Chapter 16 - Multiple Integration - 16.6 Change of Variables - Exercises - Page 905: 21

Answer

$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} xy{\rm{d}}x{\rm{d}}y = \frac{{2329}}{{12}}$

Work Step by Step

We have the mapping $G\left( {u,v} \right) = \left( {5u + 3v,u + 4v} \right)$. So, $x\left( {u,v} \right) = 5u + 3v$ and $y\left( {u,v} \right) = u + 4v$. Write $f\left( {x\left( {u,v} \right),y\left( {u,v} \right)} \right) = \left( {5u + 3v} \right)\left( {u + 4v} \right)$ $f\left( {x\left( {u,v} \right),y\left( {u,v} \right)} \right) = 5{u^2} + 23uv + 12{v^2}$ Evaluate the Jacobian of $G$: ${\rm{Jac}}\left( G \right) = \left| {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial u}}}&{\frac{{\partial x}}{{\partial v}}}\\ {\frac{{\partial y}}{{\partial u}}}&{\frac{{\partial y}}{{\partial v}}} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} 5&3\\ 1&4 \end{array}} \right| = 17$ Evaluate $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} xy{\rm{d}}x{\rm{d}}y$ over ${{\cal D}_0}$ using the General Change of Variables Formula: $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} xy{\rm{d}}x{\rm{d}}y = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal D}_0}}^{} f\left( {x\left( {u,v} \right),y\left( {u,v} \right)} \right)\left| {Jac\left( G \right)} \right|{\rm{d}}u{\rm{d}}v$ $ = 17\mathop \smallint \limits_{v = 0}^1 \mathop \smallint \limits_{u = 0}^1 \left( {5{u^2} + 23uv + 12{v^2}} \right){\rm{d}}u{\rm{d}}v$ $ = 17\mathop \smallint \limits_{v = 0}^1 \left( {\frac{5}{3}{u^3} + \frac{{23}}{2}{u^2}v + 12u{v^2}} \right)|_0^1){\rm{d}}v$ $ = 17\mathop \smallint \limits_{v = 0}^1 \left( {\frac{5}{3} + \frac{{23}}{2}v + 12{v^2}} \right){\rm{d}}v$ $ = 17\left( {\left( {\frac{5}{3}v + \frac{{23}}{4}{v^2} + 4{v^3}} \right)|_0^1} \right)$ $ = 17\left( {\frac{5}{3} + \frac{{23}}{4} + 4} \right) = \frac{{2329}}{{12}}$ Thus, $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} xy{\rm{d}}x{\rm{d}}y = \frac{{2329}}{{12}}$.
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