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Find the smallest area bounded by the cu...

Find the smallest area bounded by the curves `y=x-sin x, y= x+ cos x.`

Text Solution

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The correct Answer is:
`2sqrt(2)` sq. units
Solving given curves, we have x-sin x = x + cos x
`therefore" tan x =-1`
For smallest bounded area we must consider two consecutive solutions,
`i.e.," "x=-(pi)/(4),(3pi)/(4)`
`therefore" Required area "=|overset(3pi//4)underset(-pi//4)int[(x + cos x)-(x- sin x)]dx |`
`=|overset(3pi//4)underset(-pi//4)int[cos x + sin x] dx |`
`=[sin x - cos x ]_(-pi//4)^(3pi//4)`
`=((1)/(sqrt(2))+(1)/(sqrt(2)))-(-(1)/(sqrt(2))-(1)/(sqrt(2)))`
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