Let f: R R be a continuous function satisfying ₀^ x f(t) d t = f(x) - 1 3 - x for all x R , where is a real constant. If _ x - f(x)=12 , then the value of 12A , where A is the area of the region bounded by the curves y=f(x) , y=12 , x=0 and

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Let f: R R be a continuous function satisfying ₀^ x f(t) d t = f(x) - 1 3 - x for all x R , where is a real constant. If _ x - f(x)=12 , then the value of 12A , where A is the area of the region bounded by the curves y=f(x) , y=12 , x=0 and x=- 1 3 _ e 2 , is equal to ________.

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Correct answer: 22

Differentiating the given integral equation with respect to x using the Leibniz rule, we get: f(x) = f^ (x) 3 - f^ (x) - 3f(x) = 3 This is a linear differential equation with integrating factor e^ -3 d x = e^ -3x . Multiplying by the I.F. and integrating: f(x) e^ -3x = 3 e^ -3x d x f(x) e^ -3x = - e^ -3x + C f(x) = - + C e^ 3x Taking the limit as x - : _ x - f(x) = - = 12 = -12 To find C , we need an initial condition. Substituting x = 0 into the original integral equation: ₀⁰ f(t) d t = f(0) - 1 3 - 0 0 = f(0) - 1

Let f: R R be a continuous function satisfying ₀^ x f(t) d t = f(x) - 1 3 - x for all x R , where is a real constant. If _ x - f(x)=12 , then the value of 12A , where A is the area of the region bounded by the curves y=f(x) , y=12 , x=0 and x=- 1 3 _ e 2 , is equal to ________.

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