Question

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.

Answer 1

Answer:

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