Step 1: Solution-(1) Given: For R = [0,1] × [0,1] 2 I = √ √(xy)² cos(x³)dxdy R Our aim is to evaluate the given integral Step 2: Calculation: I = S√(xy)² cos(x³)dxdy R I = S₁₂ f₁ x² y² cos(x³) dxdy I = (f²x² cos(x³ ) d x) (f₁ y²dy) = Let x³ 3 t⇒ 3x² dx = dt f₁² fox² cos(x³) dx = cos(t)dt fox² cos(x³)dx = (sint) | So x² cos(x3)dx=(sin1 - sin0) Sox2 cos(x³)dx x² cos(x³)dx = (sin1 - 0) sin1 = 3 So y²dy = (5) 1 0 So y² dy = -0 = 1/3 Hence we get: I 3 = (sinl) (½³) = (sinl) 9

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