en
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math graphie polynomials math-format
Mean of a function
i-03-XX-1
number
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randRange(3,8) 0 a function( t ) { return [-t*t*t/3+ U/2*t*t]; } F(U)/a function( t ) { return [t, 0]; } function( t ) { return [t, mu]; } function( t ) { return [t, -t*(t-U)]; }

Consider \displaystyle \int_{0}^{U} {\color{blue}f(x)} \; dx = fractionReduce(roundTo(2,3*F(U)),3).

At what height {\color{red}h} do the two (orange) areas above and below this height match?



graphInit({ range: [[ -1, 9 ] ,[ -1, (U*U)/4 + 1 ]], scale: [ 20, 20 ], gridStep: [ 1, 18 ], tickStep: [ 2, 10 ], labelStep: [ 2, 10 ], unityLabels: true, }); label( [ 0, (U*U)/4 + 1 ], "f(x)", "above" ); label( [ 8.5, 0 ], "x", "above" ); label( [ 0, mu ], "\\color{red}h", "left" ); // draw curve style({ stroke: BLUE, strokeWidth: 2.2 }, function() { plot( function( x ) { return -(x-L)*(x-U) ; }, [ L, U ] ); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( RecTop , [0, (U/2)*0.423], 1, f); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( RecTop, [(U/2)*1.577, U], 1, f); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( f, [(U/2)*0.423, (U/2)*1.577], 1, RecTop); }); style({ stroke: RED, strokeWidth: 2.2 }, function() { plot( function( x ) { return mu ; }, [ L, U ] ); });


(F(U)-F(L))/a

The {\color{red}h} we are looking is one side of the rectangle.

graphInit({ range: [[ -1, 9 ] ,[ -1, (U*U)/4 + 1 ]], scale: [ 20, 20 ], gridStep: [ 1, 18 ], tickStep: [ 2, 10 ], labelStep: [ 2, 10 ], unityLabels: true, }); label( [ 0, (U*U)/4 + 1 ], "f(x)", "above" ); label( [ 8.5, 0 ], "x", "above" ); label( [ 0, mu ], "\\color{red}h", "left" ); // draw curve style({ stroke: BLUE, strokeWidth: 2.2 }, function() { plot( function( x ) { return -(x-L)*(x-U) ; }, [ L, U ] ); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( RecTop , [0, (U/2)*0.423], 1, f); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( RecTop, [(U/2)*1.577, U], 1, f); }); style({ fill: ORANGE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( f, [(U/2)*0.423, (U/2)*1.577], 1, RecTop); }); style({ stroke: RED, strokeWidth: 2.2 }, function() { plot( function( x ) { return mu ; }, [ L, U ] ); }); style({ fill: RED, stroke: RED, opacity: 0.2}, function() { plotParametric( RecTop, [L, U], 1, RecBot); });

The other side of the rectangle can be read on x-axis with {\color{orange}a}.

Thus the area of the rectangle is F= {\color{red}h} \cdot {\color{orange}a}.

style({ fill: BLUE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( f, [0, (U/2)*0.423], 1, RecBot); }); style({ fill: BLUE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( RecTop, [(U/2)*0.423, (U/2)*1.577], 1, RecBot); }); style({ fill: BLUE, stroke: BLUE, opacity: 0.5}, function() { plotParametric( f, [(U/2)*1.577, U], 1, RecBot); });

On the blue piece there's equality of F and the area between x-axis and parabola.

Thus the orange areas are the missing piece of both. We choose {\color{red}h} such that F equals the area between x-axis and parabola.

Therefore

\displaystyle F = {\color{red}h} \cdot {\color{orange}a} = \int_{L}^{U} {\color{blue}f(x)} \; dx = fractionReduce(roundTo(2,3*F(U)),3) and

\displaystyle {\color{red}h} = \frac 1{{\color{orange}a}} \int_{L}^{U} {\color{blue}f(x)} \; dx = \frac 1{{\color{orange}a}} \cdot fractionReduce(roundTo(2,3*(F(U)-F(L))),3) = fractionReduce(roundTo(2,3*(F(U)-F(L))),3*a).