Question

URGENT! Please Find the centroid of the area bounded by the parabola y = 4 - x2 and the x-axis.

A. (0,1.9)
B. (0,1.8)
C. (0,1.6)
D. (0,1.7)

Answer 1

We have a parabola
y=f(x)=4-x^2.

The zeroes of f(x) are located at x=-2, and x=+2 (see attached graph).

The centroid is defined as

`y0=\frac{\int_(-2)^2{f(x)*(y/2)}dx}{\int_(-2)^2{f(x)}dx}`

`=\frac{\int_(-2)^2{(4-x^2)^2/2}dx}{\int_(-2)^2{4-x^2}dx}`

`=\frac{\int_(-2)^2{(4+x^4/2-4x^2)}dx}{\int_(-2)^2{4-x^2}dx}`

`=([(8x+x^5/10-4x^3/3)]_(-2)^2)/([4x-x^3/3]_(-2)^2)`

`=([(32+32/5-64/3)])/([16-16/3])`

`=([(256/15)])/([32/3])`

`=(8)/(5)`
=1.6

Or more precisely, the centroid is at C(0,1.6)