The area bounded by the curve f(x)=x+sinx and its inverse between the ordinates x=0 to x=2π is
A
4 sq. units
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B
8 sq. units
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C
4π sq. units
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D
8π sq. units
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Solution
The correct option is B8 sq. units f(x)=x+sinx For x∈(0,π),sinx>0 ⇒f(x)>x For x∈(π,2π),sinx<0 ⇒f(x)<x
We know that, f−1(x) is the mirror image of f(x) with respect to the line y=x
So, the area between f(x) and f−1(x) is double the area between f(x) and y=x from x=0 to x=2π Which is equivalent to 4 times the area between f(x) and y=x from x=0 to x=π
So, the required area is, A=4π∫0[(x+sinx)−x]dx =4π∫0sinxdx =4[−cosx]π0 =4×2 =8sq. units