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Misc 19 - Area bounded by y-axis, y = cos x, y = sin x - Miscellaneous

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  1. Class 12
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Misc 19 The area bounded by the 𝑦-axis, 𝑦=cos⁡𝑥 and 𝑦=sin⁡𝑥 when 0≤𝑥≤ 𝜋﷮2﷯ is (A) 2 ( ﷮2 −1﷯) (B) ﷮2 −1﷯ (C) ﷮2 ﷯+1 (D) ﷮2 ﷯ Finding point of intersection B Solving 𝑦=cos⁡𝑥 and 𝑦=s𝑖𝑛⁡𝑥 cos﷮𝑥﷯=s𝑖𝑛⁡𝑥 At 𝑥= 𝜋﷮4﷯ , both are equal Also, 𝑦=cos⁡𝑥 = cos 𝜋﷮4﷯ = 1﷮ ﷮2﷯﷯ So, B = 𝜋 ﷮4﷯ , 1﷮ ﷮2﷯﷯﷯ Step 3: Finding Area Area Required = Area ABCO – Area BCO Area ABCO Area ABCO = 0﷮ 𝜋﷮4﷯﷮𝑦 𝑑𝑥﷯ Here, 𝑦= cos﷮𝑥﷯ Thus, Area ABCO = 0﷮ 𝜋﷮4﷯﷮ cos﷮𝑥﷯ 𝑑𝑥﷯ = sin﷮𝑥﷯﷯﷮0﷮ 𝜋﷮4﷯﷯ = sin﷮ 𝜋﷮4﷯− sin﷮0﷯﷯﷯ = 1﷮ ﷮2﷯﷯−0 = 1﷮ ﷮2﷯﷯ Area BCO Area BCO = 0﷮ 𝜋﷮4﷯﷮𝑦 𝑑𝑥﷯ Here, 𝑦= sin﷮𝑥﷯ Thus, Area BCO = 0﷮ 𝜋﷮4﷯﷮ sin﷮𝑥﷯ 𝑑𝑥﷯ = −c𝑜𝑠﷮𝑥﷯﷯﷮0﷮ 𝜋﷮4﷯﷯ =− cos﷮ 𝜋﷮4﷯− cos﷮ 0﷯﷯﷯﷯ =− 1﷮ ﷮2﷯﷯−1﷯ =1− 1﷮ ﷮2﷯﷯ Therefore Area Required = Area ABCO – Area BCO = 1﷮ ﷮2﷯﷯− 1− 1﷮ ﷮2﷯﷯﷯ = 1﷮ ﷮2﷯﷯+ 1﷮ ﷮2﷯﷯−1 = 2﷮ ﷮2﷯﷯−1 = ﷮2﷯−1 ∴ Option B is Correct

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