Find the centroid of the plane?

Question

Find the centroid of the plane region above the curve y=x^4, below the straight line y=x and in the first quadrant.

Answer 1

First of all, y = x⁴ and y = x intersect at two points, (0,0) and (1,1).

First calculate the area of the region.

A = ∫₀¹ (x - x⁴) dx = x²/2 - x⁵/5 |₀¹ = 1/2 - 1/5 = 3/10

Now you can get the centroid as follows.

x_c = (10/3)∫₀¹ ∫^x_x^4  x dy dx

= (10/3)∫₀¹ (xy |^x_x^4) dx

= (10/3)∫₀¹ (x² - x⁵) dx

= (10/3)(x³/3 - x⁶/6) |₀¹

= (10/3)(1/3 - 1/6)

= 5/9

and

y_c = (10/3)∫₀¹ ∫^x_x^4 y dy dx

= (10/3)∫₀¹ (y²/2) |^x_x^4 dx

= (10/3)∫₀¹ (x²/2 - x⁸/2) dx

= (10/3)(x³/6 - x⁹/18)|₀¹

= (10/3)(1/9) = 10/27

Centroid at (5/9, 10/27)