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, IX<br />
Example<br />
Determ<strong>in</strong>e the volume V of the solid E that lies above the<br />
cone z = � x 2 + y 2 <strong>and</strong> below the sphere x 2 + y 2 + z2 = z.<br />
Note:<br />
x 2 + y 2 + z 2 = z ⇐⇒ x 2 + y 2 �<br />
+ z − 1<br />
�2 =<br />
2<br />
1<br />
4 .<br />
Project E onto the xy-plane <strong>and</strong> obta<strong>in</strong> D:<br />
�<br />
D = (x, y) : x 2 + y 2 ≤ 1<br />
�<br />
4<br />
� �<br />
= (r, θ) : r ∈ 0, 1<br />
�<br />
�<br />
, θ ∈ [0, 2π] .<br />
2