Convert the Integral Below from Rectangular Coordinates to Both Cylindrical $\int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x$

question 20

Multiple Choice

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral. $\int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x$

• A) \(\frac { 729 } { 8 } \pi\)
• B) \(\frac { 6561 } { 4 } \pi\)
• C) \(6561 \pi\)
• D) \(\frac { 6561 } { 2 } \pi\)
• E) \(\frac { 6561 } { 8 } \pi\)

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Definitions:

Natural Magnets

Materials found in nature, such as magnetite, that possess magnetic properties and generate a magnetic field without any external influence.

Iron

A chemical element with symbol Fe, commonly used in its magnetic form in motors, generators, and transformers due to its ferromagnetic properties.

Cobalt

A chemical element with symbol Co and atomic number 27, known for its strength and high temperature resistance, often used in alloys and batteries.

Nonmetal Materials

Substances or materials that do not have the characteristics of metals, including aspects such as electrical conductivity, luster, or malleability.