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
