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, VII<br />
Example (cont<strong>in</strong>ued)<br />
Then:<br />
���<br />
H<br />
=<br />
=<br />
x 2 + y 2 dV<br />
� 2π<br />
0<br />
� 2π<br />
0<br />
� π<br />
2<br />
0<br />
� π<br />
2<br />
0<br />
� 1<br />
0<br />
� 1<br />
0<br />
=<br />
(ρ 2 cos 2 θ s<strong>in</strong> 2 φ + ρ 2 s<strong>in</strong> 2 θ s<strong>in</strong> 2 φ)<br />
ρ 2 s<strong>in</strong> φ dρ dφ dθ<br />
ρ 4 (cos 2 θ + s<strong>in</strong> 2 θ) s<strong>in</strong> 3 φ dρ dφ dθ<br />
� 2π<br />
0<br />
� π<br />
2<br />
0<br />
� 1<br />
0<br />
ρ 4 s<strong>in</strong> 3 φ dρ dφ dθ = · · ·