Mathematics

Let $$f:\left[ -2,3 \right] \rightarrow \left[ 0,\infty  \right) $$ be a continuous function such that $$f\left( 1-x \right) -f\left( x \right) $$ for all $$x\epsilon \left[ -2,3 \right] $$.
If $${ R }_{ 1 }$$ is the numerical value of the area of the region bounded by $$y=f\left( x \right) ,x=-2,x=3$$ and the axis of x and $${ R }_{ 2 }=\int _{ 2 }^{ 3 }{ xf\left( x \right) dx } $$, then:-


SOLUTION
$${ R }_{ 1 }=\int _{ -2 }^{ 3 }{ f\left( x \right) dx } $$
$${ R }_{ 2 }=\int _{ -2 }^{ 3 }{ xf\left( x \right) dx } $$
$$=\int _{ -2 }^{ 3 }{ \cfrac { xf\left( x \right) +\left( 1-x \right) f\left( 1-x \right)  }{ 2 } dx } $$
$$=\int _{ -2 }^{ 3 }{ \cfrac { xf\left( x \right) +(1-x)f\left( x \right)  }{ 2 } dx } $$
$$=\cfrac { 1 }{ 3 } \int _{ -2 }^{ 3 }{ f\left( x \right) dx } =\cfrac { { R }_{ 1 } }{ 2 } $$
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Subjective Medium Published on 17th 09, 2020
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