Use spherical coordinates.
24. Evaluate ∫∫∫E y2 dV, where E is the solid hemisphere x2 + y2 + z2 9, y 0.
To evaluate: The given triple integral by using spherical coordinates.
The function is .
The region B is hemisphere and .
If is a spherical region E given by , then, (1)
If is the function of and is the function of and is the function of z then, (2)
The spherical coordinates corresponding to the rectangular coordinates is,
By the given conditions, it is observed that varies from 0 to 3, varies from 0 to and varies from 0 to . Use the formula mentioned above to change the given problem into spherical coordinates. Then, by equation (1), the value of the given triple integral is,
Use (2) to integrate and apply the limit values
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