Calculate the value of the multiple integral.
31. where E is bounded by the paraboloid x = 1 − y2 − z2 and the plane x = 0
To calculate: The given triple integral.
The function is .
The region E is bounded by the paraboloid and the plane .
If is a polar rectangle R given by where , then, (1)
If is the function of and is the function of then,
Substitute in the given equation as shown below.
From the given conditions, x varies from 0 to , y varies from to 1 and z varies from to . First compute the integral with respect to z and apply the limit.
Since the value to be integrated and the limit of z are difficult, use of polar coordinates makes the problem easier. Substitute
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