Question

Please help NO LINKS

Answer 1

`\bar{x} = 0`

`\bar{y} =(136)/(125)`

Explanation:

Let's define our functions
`f(x)\:\text{and}\:g(x)` as follows:

`f(x) = x^2 + 1`

`g(x) = 6x^2`

The two functions intersect when
`f(x)=g(x)` and that occurs at
`x = \pm(1)/(5)` so they're going to be the limits of integration. To solve for the coordinates of the centroid
`\bar{x}\:\text{and}\:\bar{y}`, we need to solve for the area A first:

`\displaystyle A = \int_a^b [f(x) - g(x)]dx`

`\displaystyle \:\:\:\:\:\:\:=\int_{-(1)/(5)}^{+(1)/(5)}[(x^2 + 1) - 6x^2]dx`

`\displaystyle \:\:\:\:\:\:\:=\int_{-(1)/(5)}^{+(1)/(5)}(1 - 5x^2)dx`

`\displaystyle \:\:\:\:\:\:\:=\left(x - (5)/(3)x^3 \right)_{-(1)/(5)}^{+(1)/(5)}`

`\:\:\:\:\:\:\:= (28)/(75)`

The x-coordinate of the centroid
`\bar{x}` is given by

`\displaystyle \bar{x} = (1)/(A)\int_a^b x[f(x) - g(x)]dx`

`\displaystyle \:\:\:\:\:\:\:= (75)/(28)\int_{-(1)/(5)}^{+(1)/(5)} (x - 5x^3)dx`

`\:\:\:\:\:\:\:=(75)/(28)\left((1)/(2)x^2 -(5)/(4)x^4 \right)_{-(1)/(5)}^{+(1)/(5)}`

`\:\:\:\:\:\:\:= 0`

The y-coordinate of the centroid
`\bar{y}` is given by

`\displaystyle \bar{y} = (1)/(A)\int_a^b (1)/(2)[f^2(x) - g^2(x)]dx`

`\displaystyle \:\:\:\:\:\:\:=(75)/(28)\int_{-(1)/(5)}^{+(1)/(5)} (1)/(2)(-35x^4 + 2x^2 + 1)dx`

`\:\:\:\:\:\:\:=(75)/(56) \left[-7x^5 + (2)/(3)x^3 + x \right]_{-(1)/(5)}^{+(1)/(5)}`

`\:\:\:\:\:\:\:=(136)/(125)`