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, XII<br />
Example (cont<strong>in</strong>ued)<br />
On the side:<br />
<strong>and</strong><br />
� 1<br />
2<br />
0<br />
r<br />
� 1<br />
2<br />
0<br />
r<br />
2 − r 2 dr =<br />
r 2<br />
4<br />
� 1<br />
4 − r 2 dr = − 1<br />
2<br />
− r 3<br />
3<br />
� 0<br />
1<br />
4<br />
= 1<br />
2<br />
�<br />
�<br />
�<br />
�<br />
r= 1<br />
2<br />
r=0<br />
√ u du<br />
� 1<br />
4<br />
0<br />
= 1 1 1<br />
− =<br />
16 24 48<br />
√ u du = u 3<br />
2<br />
3<br />
�<br />
�<br />
�<br />
�<br />
�<br />
u= 1<br />
4<br />
u=0<br />
= 1<br />
24 .