Consider the region D in the first octant that is inside both of the sphere x^2+y^2 + z^2 =...
Question:
Consider the region {eq}D {/eq} in the first octant that is inside both of the sphere {eq}x^2+y^2 + z^2 = 2^2 {/eq} and the cylinder {eq}x^2+y^2 =1 {/eq}. Set up a triple integral to find the volume of the solid in
a. rectangular coordinates
b. cylindrical coordinates
c. spherical coordinates
Calculating Volume using Rectangular, Cylindrical, and Spherical Coordinates:
If we wish to find the volume of a region {eq}D {/eq} in three dimensions, we can use a triple integral. Depending on the coordinate system we use, we set up the integrals with different limits of integration. The volume integral is {eq}V = \displaystyle\iiint_D dV. {/eq}
In rectangular coordinates we use the differential {eq}dV = dz \: dy \: dx. {/eq}
In cylindrical coordinates we use the differential {eq}dV = r \: dz \: dr \: d\theta. {/eq}
In spherical coordinates we use the differential {eq}dV = \rho^2 \sin \phi \: d\rho \: d\phi \: d\theta. {/eq}
Answer and Explanation: 1
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View this answerWe are given the region {eq}D {/eq} in the first octant that is inside both of the sphere {eq}x^2+y^2 + z^2 = 2^2 {/eq} and the cylinder {eq}x^2+y^2...
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How to Find Volumes of Revolution With Integration
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Chapter 14 / Lesson 5
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The volume of a revolution can be calculated using the slicing method, the disk method, and the washer method. Explore the processes of the three methods and discover how to use them to find the volumes of revolution.
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