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Section 16-5: Exercise # 5

> restart;

> with( plots ):

Warning, the name changecoords has been redefined

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We begin by plotting the region S.

> P1 := plot( 0, x=0..1 ):

> P2 := plot( exp(-x), x=0..1 ):

> P3 := plot( [0,y, y=0..1] ):

> P4 := plot( [1,y, y=0..exp(-1)] ):

> plotS := display( [P1,P2,P3,P4] ):

> plotS;

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The density function is

> delta := 1;

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The iterated integral for the mass is

> M := Int( Int( delta, y=0..exp(-x) ), x=0..1 );

with value

> value( M ) = evalf( M );

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The moment about the y axis (x=0) is the iterated integral

> M_y := Int( Int( x*delta, y=0..exp(-x) ), x=0..1 );

with value

> value( M_y ) = evalf( M_y );

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The moment about the x axis (y=0) is the iterated integral

> M_x := Int( Int( y*delta, y=0..exp(-x) ), x=0..1 );

> value( M_x ) = evalf( M_x );

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The center of mass is

> xbar := value( M_y/M ):

> ybar := value( M_x/M ):

> CoM := simplify([xbar,ybar]);

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or, as floating point numbers:

> evalf(CoM);

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To locate the center of mass on the lamina, plot the center of mass and add it to the plot of S:

> plotCM := pointplot( [xbar,ybar] ):

> display( [plotS,plotCM] );

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This location for the center of mass is reasonable because the density is much higher towards the top of the region. For the sake of illustration, if the density function is ( i.e. , a uniform density) then the center of mass is (0.41802,0.34197) and if then the center of mass is (0.34348,0.48842).

> CoM0 := [ 0.41802, 0.34197 ]:

> CoM1 := [ 0.34348, 0.48842 ]:

> plotCM1 := pointplot( [ CoM0, CoM1 ], color=green ):

> display( [plotS,plotCM,plotCM1] );

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