Find the Mass and Center of Mass of a Thin $x ^ { 2 } + y ^ { 2 } = r ^ { 2 }$

question 58

Essay

Find the mass and center of mass of a thin wire in the shape of a quarter-circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 }$ , $x \geq 0 , y \geq 0$ , if the density function is $\rho ( x , y ) = x + y$ .

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Definitions:

Marine Sediments

Deposits accumulated on the ocean floor, derived from chemical precipitates, organic matter, or the erosion of terrestrial materials.

Black Smokers

Hydrothermal vents on the seafloor in which hot water from within the rock jets out into cold seawater and forms a cloud of minerals, especially those rich in sulfur.

Sulfide Minerals

Minerals composed of sulfide ions and one or more metals; important as ore minerals and for their chemical properties.

Fast-Spreading Ridge

A geological feature where tectonic plates are diverging at a rapid rate, leading to significant volcanic activity and the formation of new oceanic crust.