Find the volume of the solid region bounded above by the sphere x^2 + y^2 + z^2 = 6 and below by...


Find the volume of the solid region bounded above by the sphere {eq}x^2 + y^2 + z^2 = 6 {/eq} and below by the paraboloid {eq}x^2 + y^2 = z {/eq}.

Calculating the Volume of a Solid Using Cylindrical Coordinates

In order to calculate the volume of a three dimensional solid {eq}R {/eq} using cylindrical coordinates we use the following formula for {eq}dV: {/eq}

{eq}dV = r \: dr \: d\theta \: dz {/eq}

which can be rearranged for alternate orders of integration. Next we need to describe the region {eq}R {/eq} in terms of cylindrical coordinates. Suppose {eq}z = f(x, y) {/eq} describes the lower boundary of the solid and {eq}z = g(x, y) {/eq} describes the upper boundary. Then we rewrite each of the functions in terms of {eq}r {/eq} and {eq}\theta {/eq} using the identities

{eq}x = r \cos \theta \\ y = r \sin \theta {/eq}

Next determine the projection of the solid into the {eq}xy {/eq} plane by setting these two functions equal to each other and simplifying. We can describe this projection in polar coordinates, which then gives us the limits of integration for {eq}r {/eq} and {eq}\theta. {/eq} Finally we write the volume integral as

{eq}V = \displaystyle\iiint_R r \: dz \: dr \: d\theta {/eq} and evaluate the triple integral iteratively.

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The sphere can be described in cylindrical coordinates as follows:

{eq}\begin{eqnarray*}z^2 & = & 6 - x^2 - y^2 \\ z^2 & = & 6 - r^2 \\ z & = &...

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Double Integrals: Applications & Examples


Chapter 12 / Lesson 14

In mathematics, double integrals enable the process of integration in two-dimension areas. Explore the applications and examples of double integrals. Review the background on integrals, finding the area of a bounded region, the ordering of integration, finding a volume under the surface, and calculating the mass.

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