Question

The area of the region A = \({(x,y):|cos⁡x−sin⁡x|≤y≤sin⁡x,0≤x≤\frac{π}{ 2} }\) is

• A. \(\sqrt 5−2\sqrt 2+1\)
• B. \(1-\frac{3}{\sqrt 2}+\frac{4}{\sqrt 5}\)
• C. \(\frac{3}{\sqrt 5}-\frac{3}{\sqrt 2}+1\)
• D. \(\sqrt 5+2\sqrt 2−4.5\)

Answer 1

The Correct Option is A

Step 1: The intersection point of cos x - sin x = sin x is given by:

tan x = 1/2

So, let ψ = tan^-1(1/2)

Hence, sin ψ = 1/√5 and cos ψ = 2/√5

Step 2: The graph of the equation is shown below:

Step 3: Now, the area between the curves can be calculated using the integral:

Area = ∫_ψ^π/2 (sin x - |cos x - sin x|) dx 

Step 4: Break the integral into two parts:

= ∫_ψ^π/4 (sin x - (cos x - sin x)) dx + ∫_π/4^π/2 (sin x - (sin x - cos x)) dx 

Step 5: Simplify the integrals:

= ∫_ψ^π/4 (2 sin x - cos x) dx + ∫_π/4^π/2 cos x dx 

Step 6: Now, evaluate the integrals:

= [-2 cos x - sin x]_ψ^π/4 + [sin x]_π/4^π/2 

Step 7: Substituting the limits:

= -√2 ¹/_√2 + 2 cos ψ + sin ψ + (1 - 1/√2) 

Step 8: Final evaluation:

= -√2 ¹/_√2 + 2 (2/√5) + (1/√5) + (1 - 1/√2) 

Conclusion: Therefore, the area is:

√5 - 2√2 + 1