1.

The area enclosed by the curves`y= sinx+cosx and y = | cosx-sin x |` over the interval `[0,pi/2]`A. `4(sqrt(2)-1)`B. `2sqrt(2)(sqrt(2)-1)`C. `2(sqrt(2)+1)`D. `2sqrt(2)(sqrt(2)+1)`

Answer» Correct Answer - b
Let A denote the area enclosed by the given curves. Then,
`A=underset(0)overset(pi//2)int |(sinx + cos x )-|cos x - sin x ||dx`
`because |cos x - sin x |={{:(cos x -sinx ","0 le x le (pi)/(4)),(-(cos x-sin x)","(pi)/(4) le x le (pi)/(2)):}`
`therefore A=underset(0)overset(pi//4)int |(sin x + cosx )-(cosx-sinx)|dx+underset(pi//4)overset(pi//2)int |(sin x + cos x )+(cos x - sin x)|dx`
`implies A =underset(0)overset(pi//4)int |2sin x|dx+ underset(pi//4)overset(pi//2)int |2cos x|dx`
`implies A=2underset(0)overset(pi//4)int sin x dx + 2 underset(pi//4)overset(pi//2)int cos x dx`
`implies A =[-2cos x]_(0)^(pi//4)+[2sinx ]_(pi//4)^(pi//2)`
`implies A=(-2cos pi//4 + 2) + (2sin pi//2-2sin pi//4)`
`implies A=(2-sqrt(2))+(2-sqrt(2))=2(2-sqrt(2))=2(2-sqrt(2))=2sqrt(2)(sqrt(2)-1)`


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