These curves define several areas, but I see there is only one area that is bounded by all 4 curves, so I assume this is the area whose centroid you want.

The curves are:

y = - x^2
y = x - 2
y = - x - 2
y = 2

If the area = A then

the x-coordinate of the centroid = (integral over A) x dA/A
and the y-coordinate = (integral over A) y dA/A

Putting - x in place of x leaves the equations unchanged (the 2nd and 3rd equations swap places) so it follows that the area is symmetrical about the y-axis. Thus the x-coordinate of the centroid = 0.

The area is bounded below by:

y = - x - 2 on x E [- 4, - 1]
y = - x^2 on x E [- 1, 1]
y = x - 2 on x E [1, 4].

The area is bounded above by y = 2 on x E [- 4, 4].

Thus A = 2 integral (- 4 —-> - 1) [2 - (- x - 2)] dx + integral (- 1 —-> 1) [2 - (- x^2)] dx
= 2 integral (- 4 —-> - 1) (x + 4) dx + integral (- 1 —-> 1) (x^2 + 2) dx
= 2 (x^2/2 + 4x) | (x = - 4 —-> - 1) + (x^3/3 + 2x) | (x = - 1 —-> 1)
= 2[(1/2 - 4) - (8 - 16)] + (1/3 + 2) - (- 1/3 - 2)
= 2(- 7/2 + 8) + 4 + 2/3
= 9 + 4 + 2/3
= 13 2/3 or 41/3

And the y=coordinate of the centroid =

(3/41) (integral over A) y dA

= (3/41) [integral (x = - 4 —-> - 1) integral (y = - x - 2 —-> 2) y dy dx
+ integral (x = - 1 —-> 1) integral (y = - x^2 —-> 2) y dy dx
+ integral (x = 1 —-> 4) integral (y = x - 2 —-> 2) y dy dx]

= (3/41) {integral (x = - 4 —-> - 1) [2^2/2 - (x + 2)^2/2] dx + integral (x = - 1 —-> 1) (2^2/2 - x^4/2) dx + integral (x = 1 —-> 4) [2^2/2 - (x - 2)^2/2] dx}

= (3/41) {[2x - (x + 2)^3/6] | (x = - 4 —-> - 1) + (2x - x^5/10) | (x = - 1 —-> 1) + [2x - (x - 2)^3/6] | (x = 1 —-> 4)}

= (3/41) [(-2 - 1/6) - (- 8 + 8/6) + (2 - 1/10) - (- 2 + 1/10 ) + (8 - 8/6) - (2 + 1/6)]
= (3/41)(- 2 - 1/6 + 8 - 4/3 + 2 - 1/10 + 2 - 1/10 + 8 - 4/3 - 2 - 1/6)
= (3/41)(16 - 2/6 - 8/3 - 2/10)
= (3/41)(480 - 10 - 80 - 6)/30
= (3 * 384)/(41 * 30)
= 384/410
= 192/205

Thus the centroid of the area = (0, 192/205).

FML

maiabear:

hisjuliette:

What`s next in this sequence:

1, 2, 6, 42, 1806, _________ ?

As I can remember, Engineers can solve this in 3 minutes, Architects in 3 hours, Doctors in 3 days, and so on. :))

I know, this is so last year. :P

Waaa! Ano sagot dito? Hahahahahaha!

3,263,442. Hehe. ;)

What`s next in this sequence:

1, 2, 6, 42, 1806, _________ ?

As I can remember, Engineers can solve this in 3 minutes, Architects in 3 hours, Doctors in 3 days, and so on. :))

I know, this is so last year. :P