Find the centroid of the region enclosed by y = x³ - 2 and y = 3x + 7 from x = -1 to x = 0. A) (-0.477419, 1.668203) B) (0.425806, 1.840246) C) (1.840246, -0.477419) D) (-0.477419, 1.840246) E) (0.425806, 1.668203)

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Quizplus · Accepted Answer

To find the centroid of the region, we need to use the following formulas: x-bar = (1/A) * ∫[a,b] (x*f(x)) dx y-bar = (1/A) * ∫[a,b] ((1/2)*f(x)^2) dx where A is the area of the region, f(x) is the difference between the two functions, and [a,b] is the interval of integration. First, we need to find the intersection point of the two functions: x³ = 3x + 7 x = -2.333333 Since we're only interested in the region from x = -1 to x = 0, we can ignore the intersection point. Next, we can calculate the area of the region using the following formula: A = ∫[-1,0] (f(x)) dx f(x) = (x³ - 2) - (3x + 7) f(x) = -2x - 9 A = ∫[-1,0] (-2x - 9) dx A = [-x^2 - 9x] from -1 to 0 A = 8/3 Now we can use the formulas for x-bar and y-bar: x-bar = (1/A) * ∫[-1,0] (x*f(x)) dx x*f(x) = x*(-2x - 9) = -2x^2 - 9x x-bar = (1/(8/3)) * ∫[-1,0] (-2x^2 - 9x) dx x-bar = (-8/3) * [-x^3/3 - (9/2)*x^2] from -1 to 0 x-bar = (-8/3) * [(1/3) - (9/2)] x-bar = -0.477419 y-bar = (1/A) * ∫[-1,0] ((1/2)*f(x)^2) dx (1/2)*f(x)^2 = (1/2)*(-2x - 9)^2 = 2x^2 + 9x + 20.25/2 y-bar = (1/(8/3)) * ∫[-1,0] (2x^2 + 9x + 20.25/2) dx y-bar = (8/3) * [(2/3)*x^3 + (9/2)*x^2 + (20.25/2)*x] from -1 to 0 y-bar = (8/3) * [(2/3) - (9/2)*(-1) + (20.25/2)] y-bar = 1.668203 Therefore, the centroid of the region is (-0.477419, 1.668203), which corresponds to choice A.

Find the Centroid of the Region Enclosed by Y =