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.

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.    

Answer 1

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) = 0.65

Average 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.