Transcription of Mass, Centers of Mass, and Double Integrals
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Mass, Centers of Mass, and Double Integrals
Mass, Centers of Mass, and Double IntegralsSuppose a 2-D regionRhas density (x, y)at each point(x, y). We can partitionRinto subrectangles,withmof them in thex-direction, andnin they-direction. Suppose each subrectangle has width xand height y. Then a subrectangle containing the point( x, y)has approximate mass ( x, y) x yand the mass ofRis approximatelym i=1n j=1 (xi, yi) x ywhere(xi, yi)is a point in thei, j-th subrectangle. Lettingmandngo to infinity, we haveM=mass ofR= R (x, y) , the moment with respect to thexaxis can be calculated asMx= Ry (x, y)dAand the moment with respect to theyaxis can be calculated asMy= Rx (x, y) we may calculate the center of mass ofRviacenter of mass ofR= ( x, y) =(MyM,MxM).
We can partition R into subrectangles, with m of them in the x-direction, and n in the y-direction. Suppose each subrectangle has width ∆x and height ∆y. Then a subrectangle containing the point (ˆx,yˆ) has approximate mass ... function ρ(x,y) = 1 y +1 so that R is denser near the x-axis. As a result, we would expect the center of mass ...
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