Find the centroid of the thin plate bounded by the graphs of the functions g (x) = x^2 (x - 1) + 2 and f (x) = x^2 + 2 with delta = 4 and M=mass of the region covered by the plate.

X = 6/5

Y = 176/56

Step-by-step explanation:

Given

g (x) = x² (x-1) + 2

f (x) = x²+2

We find the limits as follows: x² (x-1) + 2 = x²+2

⇒ x² (x - 1 - 1) = x² (x - 2) = 0

⇒ x₁ = 0 and x₂ = 2

If x = 1

g (1) = (1) ² (1 - 1) + 2 = 2

f (x) = (1) ² + 2 = 3

then

f (x) > g (x)

We get the Area

A = ∫ (f (x) - g (x)) dx = ∫ ((x²+2) - (x² (x-1) + 2)) dx = ∫x² (2-x) dx = (2/3) x³ - (1/4) x⁴+C

A = (16/3) - 4 = 4/3

X = (1/A) ∫ (x (f (x) - g (x))) dx = (3/4) ∫ (x³ (2-x)) dx = (3/4) ((1/2) x⁴ - (1/5) x⁵) + C

X = (3/4) ((1/2) (2) ⁴ - (1/5) (2) ⁵) = (3/4) (8-32/5) = (3/4) (8/5) = 6/5

Y = (1 / (2A)) ∫ ((f (x)) ² - (g (x)) ²) dx = (3/8) ∫ ((x²+2) ² - (x² (x-1) + 2) ²) dx

Y = (3/8) ∫ (-x⁶+2x⁵-4x³+8x²) dx = (3/8) ( - (1/7) x⁷ + (1/3) x⁶-x⁴ + (8/3) x³+C)

Y = (3/8) ( - (1/7) (2) ⁷ + (1/3) (2) ⁶ - (2) ⁴ + (8/3) (2) ³) = (3/8) (176/21) = 176/56