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Question: A plane region is bounded by the curve y = x2 , y = 4x , x = 0 and x = 1. a) Find the area of the plane. b) Calculate the centroid of the plane.
A plane region is bounded by the curve y  =  x2 ,  y  =  4x ,  x  =  0  and x  =  1.a)         Find the area of the plane.b)         Calculate the centroid of the plane.    

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Answered By:

Shin C.
K-12 Math, Calculus, and SAT Math I and II Expert
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Written Explanation:

Part B)We will find the center for x and y separately.First, average value of x for centroids is defined as x average = ( 1 / (total area) ) * ∫(0, 1, x * (4x - x^2) , dx).= ( 1 / 5/3 ) * ∫ (0, 1, 4x^2 - x^3 , dx) = 0.65Average value of y for centroids is defined as y average = ( 1 / (2 * total area) ) *∫ (0, 1, (4x)^2 - (x^2)^2 , dx).= ( 1 / 10/3 ) * ∫(0, 1, 16x^2 - x^4 , dx) = 1.54.
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