find the centroid of the curves 2x+y=6,x=0,y=0?

Answer 1

Since your area is a right triangle, the centroid is just 1/3 the way from the origin to ech intercept. The intercepts are found by setting x and y equal to zero separately.

2*0 + y = 6 gives the intercept (0, 6)
2x + 0 = 6 gives (3, 0)

The centroid then is: (1, 2)

You, of course can do it by integration as well, (use S for integral):

ybar = S(y dA)/S(dA)
Limits on y are 0 to 6, dA = x dy = ((6-y)/2) dy

xbar = S(x dA)/S(dA)
Limits on x are 0 to 3, dA = y dx = (6 - 2x) dx

Answer 2

A = 6*3/2
y = -2x + 6
dM = x(-2x + 6)dx
M = -x^2 + 6x (0→x→3) = -9 + 18 = 9
Xc = 1

x = (1/2)(6 - y)
dM = y(1/2)(6 - y)dy
M = 3y - (1/4)y^2 (0→y→6) = 18 - 36/4 = 9
Yc = 1
(Xc,Yc) = (1, 1)

Answer 3

What!?