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 906: 42

Answer

We derive formula (5) in Section 16.4: $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$ $ = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{r = {r_1}\left( \theta \right)}^{{r_2}\left( \theta \right)} \mathop \smallint \limits_{z = {z_1}\left( {r,\theta } \right)}^{{z_2}\left( {r,\theta } \right)} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $

Work Step by Step

In cylindrical coordinates, the mapping from the region ${{\cal W}_0}$ in $\left( {r,\theta ,z} \right)$-space to the region ${\cal W}$ in $\left( {x,y,z} \right)$-space is given by $x = r\cos \theta $, ${\ \ \ \ }$ $y = r\sin \theta $, ${\ \ \ \ }$ $z=z$ Let the region description of ${{\cal W}_0}$ be given by ${{\cal W}_0} = \left\{ {\left( {r,\theta ,z} \right)|{r_1}\left( \theta \right) \le r \le {r_2}\left( \theta \right),{\theta _1} \le \theta \le {\theta _2},{z_1}\left( {r,\theta } \right) \le z \le {z_2}\left( {r,\theta } \right)} \right\}$ Write the mapping: $G\left( {r,\theta ,z} \right) = \left( {r\cos \theta ,r\sin \theta ,z} \right)$ Evaluate the Jacobian of $G$: ${\rm{Jac}}\left( G \right) = \left| {\begin{array}{*{20}{c}} {\frac{{\partial x}}{{\partial r}}}&{\frac{{\partial x}}{{\partial \theta }}}&{\frac{{\partial x}}{{\partial z}}}\\ {\frac{{\partial y}}{{\partial r}}}&{\frac{{\partial y}}{{\partial \theta }}}&{\frac{{\partial y}}{{\partial z}}}\\ {\frac{{\partial z}}{{\partial r}}}&{\frac{{\partial z}}{{\partial \theta }}}&{\frac{{\partial z}}{{\partial z}}} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} {\cos \theta }&{ - r\sin \theta }&0\\ {\sin \theta }&{r\cos \theta }&0\\ 0&0&1 \end{array}} \right|$ $ = \cos \theta \left( {r\cos \theta } \right) - \left( { - r\sin \theta } \right)\sin \theta = r{\cos ^2}\theta + r{\sin ^2}\theta $ So, ${\rm{Jac}}\left( G \right) = r$. Using the general Change of Variables Formula, Eq. (16), we get $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$ $ = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal W}_0}}^{} f\left( {x\left( {r,\theta ,z} \right),y\left( {r,\theta ,z} \right),z\left( {r,\theta ,z} \right)} \right)\left| {Jac\left( G \right)} \right|{\rm{d}}r{\rm{d}}\theta {\rm{d}}z$ $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$ $ = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{r = {r_1}\left( \theta \right)}^{{r_2}\left( \theta \right)} \mathop \smallint \limits_{z = {z_1}\left( {r,\theta } \right)}^{{z_2}\left( {r,\theta } \right)} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $ The last integral is formula (5) in Section 16.4.
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