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}.
Answer and Explanation: 1
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View this answerThe 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...
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Volumes of Shapes: Definition & Examples
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Chapter 11 / Lesson 9
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In this lesson, learn the definition of volume and how to find the volume of objects of various shapes. Learn from various solved volume examples.
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