Find the area of the shaded region bounded by y = 9 x and y = x(sqrt(24^2-x^2))

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Explanation / Answer

a) Draw a graph of the functions. In the range 0 to 0.2387 , e^-3x > sqrt(x) and in the region 0.2387 to 1, sqrt(x) > e^(-3x) The two curves intersect at x=0.23873413 Area = ? [e^(-3x) - x^(1/2) ] dx , limits [0, 0.23873413] + ? [x^(1/2) - e^(-3x)] dx, limit [ 0.23873413,1] = -(1/3) e^-3x - (2/3) x^(3/2) limits [0, 0.23873413] = -0.240632 - (-0.33333) = 0.092698 plus (2/3) x^(3/2) + (1/3) e^(-3x) limits [ 0.23873413, 1] = 0.6832624 - (0.2406324) =0.44263 Area =0.092698 + 0.44263 =0.535328 b) Volume = p ? (1-sqrt(x))^2 - (1-e^(-3x))^2 dx , limit 0 to 1 Volume= p ? (1-2x^(1/2)+x) - (1-2e^(-3x)+e^(-6x) dx You can complete the integration

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