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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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