Y=4-x, the x-axis, the y-axis in the first quadrant. Find the centroid. (using definite integration).?

Question

Y=4-x, the x-axis, the y-axis in the first quadrant. Find the centroid. (using definite integration).?

Answer 1

The figure is a right triangle with sides 4, 4, and 4 sqrt(2)

area = 8

Since the figure is symmetric about a line x=y, the centroid will lie along that line.

So one only has to integrate once to find the centroid:

Integrate 'x'  from 0 to 4

integral of (4-x)x dx from u=0 to u=4

= 2x^2 - 1/3 x^3 from 0 to 4 = 32/3

location of centroid along x = 32/3  / area = 4/3

due to symmetry the location of centoid is also = 4/3

centroid = (4/3, 4/3)