Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
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MATH 209—<br />
Calculus,<br />
III<br />
Volker Runde<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
cyl<strong>in</strong>drical<br />
coord<strong>in</strong>ates<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
<strong>spherical</strong><br />
coord<strong>in</strong>ates<br />
Examples, II<br />
Example (cont<strong>in</strong>ued)<br />
Thus:<br />
���<br />
m = C<br />
= C<br />
E<br />
� 2π � 1<br />
0<br />
0<br />
� x 2 + y 2 dV = C<br />
� 2π � 1 � 4<br />
r 2 (4 − (1 − r 2 )) dr dθ = 2Cπ<br />
�<br />
= 2Cπ r 3 �<br />
r 5 �<br />
+ �<br />
5 �<br />
0<br />
0<br />
1−r 2<br />
� 1<br />
0<br />
r=1<br />
r=0<br />
r 2 dz dr dθ<br />
3r 2 + r 4 dr<br />
�<br />
= 12Cπ<br />
.<br />
5