The coordinates of the centroid for f(x) = x "for all" epislon[0,2] are (4/3, 2/3); how far away is the centroid for the planner figure governed by g(x) = 3x^2 along the same interval

a. (3/2, 18/5)

b. Exactly 3.9 units

c. 2.9 units

d. 1.5 units

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- Jul 15th 2008, 01:34 PMshanniepooh2Coordinates of a centroid
The coordinates of the centroid for f(x) = x "for all" epislon[0,2] are (4/3, 2/3); how far away is the centroid for the planner figure governed by g(x) = 3x^2 along the same interval

a. (3/2, 18/5)

b. Exactly 3.9 units

c. 2.9 units

d. 1.5 units - Jul 17th 2008, 09:48 PMChris L T521
The x coordinate of the centroid of $\displaystyle g(x)=3x^2 \ \forall \ \varepsilon[0,2]$ is:

$\displaystyle \bar x=\frac{1}{\int_0^2 3x^2\,dx}\int_0^2\int_0^{3x^2}x\,dy\,dx=\frac{1}{8 }\int_0^2\left.\bigg[xy\bigg]\right|_{y=0}^{3x^2}\,dx$

$\displaystyle =\frac{1}{8}\int_0^2 3x^3\,dx=\frac{3}{32}\left.\bigg[x^4\bigg]\right|_0^2=\color{red}\boxed{\frac{3}{2}}$

From how I interpreted the problem, D appears to be the answer...

Does this make sense?

--Chris