Locate the centroid x and y of a shape bounded by y x and y

Locate the centroid x and y of a shape bounded by y = x and y = x^3/4. Locate the centroid y of the shape shown below.

Solution

The centroid is obtained by integrating the function (f1-f2)*x for the xcoordinate and y*( g1-g2) dy where f(x) and g(y) are the given functions and the inverse functions.

The integral are to be divided by the areas gotten by integrating f1(x) -f2(x) over x and the limits are 0 to 4 ( the intersections of the two curves)

these results are Area =2.667

X= 5.333/2.667

= 2

Y = 4.27/2.67 = 1.6

For the second part do trhe same for the parabola and find the centroid, then find the centroid of the rectangle, the overall centroid is gotten by takiing moments.