Evaluate the triple integral \iiint_{E} (x + y + z) dV, where E is the solid in the first octant...

Question:

Evaluate the triple integral

{eq}\displaystyle\; \iiint_{E} \left(x + y + z\right) \,dV {/eq},

where {eq}E\, {/eq} is the solid in the first octant that lies under the paraboloid {eq}\displaystyle\; z = 4 - x^{2} - y^{2} {/eq}.

Evaluation of Triple Integrals:

To evaluate the triple integral we need to determine first the limits fro integration from the given bounding equation. Here, the given region is circular thus it is easier to integrate when using the cylindrical coordinates which to convert from the rectangular to cylindrical the coversion formulas are the following {eq}\displaystyle r^{2}=x^{2}+y^{2},\:x=r\cos \theta ,\:y=r\sin \theta {/eq}.

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The limits for integrations are,

{eq}\displaystyle 0\leq \theta \leq \frac{\pi }{2},\:0\leq r\leq 2,\:0\leq z\leq 4-r^{2} {/eq}

Thus,

{eq}\display...