Finding the y-centroid of y=x^3 between the x-axis and x=2

[t.kirschner99]

Homework Statement

Find the y-centroid of y=x³ between the x-axis, x=2, where area is 4.

Homework Equations

y_c = integral of (ydA/A)
dA = g(y)dy
y_c = h/2 * (n+1)/(2n+1)

The Attempt at a Solution

g(y) = y^1/3
A = 4
y_c = integral of (y*y^1/3/4 dy)
= integral(y^4/3 * 1/4 dy)
= 3/7 y^7/3 * 1/4
Now since the upper limit is y=8, and lower is y=0, you evaluate this expression between 0 and 8.
= 3/28 * (8)^7/3
= 13.71
So that is the answer I get, but another formula I have is the one using h and n above where h = upper limit = 8, and n = order = 3.
With that, I get the answer of 2.28 which is the answer in the answer key.

My question is, where am I going wrong in the integral?

Thank you for your help!

 

[LCKurtz]

Homework Statement

Find the y-centroid of y=x³ between the x-axis, x=2, where area is 4.

Homework Equations

y_c = integral of (ydA/A)
dA = g(y)dy
y_c = h/2 * (n+1)/(2n+1)

The Attempt at a Solution

g(y) = y^1/3
A = 4
y_c = integral of (y*y^1/3/4 dy)

Your $y_c$ is not set up correctly. Set up correctly as a double integral it should look like this:$$
\frac 1 4 \int_0^8\int_{y^{\frac 1 3}}^2 y~dxdy$$In other words, your $y$ integrand should be $y(2-y^{\frac 1 3})$.

 

[Mark44]

Find the y-centroid of y=x³ between the x-axis, x=2, where area is 4.

This problem statement is somewhat vague. A better problem statement would be the following.

Find the y coordinate of the centroid of the region bounded by the curve y = x³, the x-axis, and the lines x = 0 and x = 2.