Find the volume of the solid bounded above by the sphere x^2 + y^2 + z^2 = 64 and below by the...
Question:
Find the volume of the solid bounded above by the sphere {eq}x^2 + y^2 + z^2 = 64 {/eq} and below by the cone {eq}z = \sqrt{x^2 + y^2} {/eq}.
Cylindrical Coordinates:
The problem above is best handled using some form of polar coordinates: either cylindrical or spherical. Because they are typically easier to work with, we will be using cylindrical coordinates. Recall
{eq}x = r \cos \theta {/eq}
{eq}y = r \sin \theta {/eq}
{eq}z = z {/eq}
{eq}r^2 = x^2+y^2 {/eq}
{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}
{eq}dV = r \ dz \ dr \ d\theta {/eq}
Answer and Explanation: 1
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View this answerIn cylindrical coordinates we have {eq}r \leq z \leq \sqrt{8-r^2} {/eq}. Next, {eq}r {/eq} is bounded by the intersection of the surfaces:
{eq}\beg...
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