I with this finding a centroid of a triangle

[rock.freak667]

Homework Statement

Well I just need to understand how to find the centroid of a triangle, I know it's 2/3 from the vertex, but I need to know how finding it is done.

Homework Equations

$$\overline{x}= \frac{\int x dA}{\int dA}$$

$$\overline{y}=\frac{\int y dA}{\int dA}$$

The Attempt at a Solution

Firstly I drew a triangle using the equation y=hx/b.



Then I considered a small rectangular element, whose height is and width is $\delta x$.

the area of this element is

The Area of this small element is $\delta A=y \delta x$

Now the sum of all the infinitesmal areas is given by

$$dA=\sum_{x=0}^{x=b} y \delta x$$

as $\delta x \rightarrow 0$

$$\int dA=\int_0 ^b y dx$$

Homework Statement

Well I just need to understand how to find the centroid of a triangle, I know it's 2/3 from the vertex, but I need to know how finding it is done.

Homework Equations

$$\overline{x}= \frac{\int x dA}{\int dA}$$

$$\overline{y}=\frac{\int y dA}{\int dA}$$

The Attempt at a Solution

Firstly I drew a triangle using the equation y=hx/b.



Then I considered a small rectangular element, whose height is and width is $\delta x$.

the area of this element is

The Area of this small element is $\delta A=y \delta x$

Now the sum of all the infinitesmal areas is given by

$$dA=\sum_{x=0} ^{x=b} y \delta x$$

as $\delta x \rightarrow 0$

$$\int dA=\int_0 ^b y dx$$

So the x-coordinate of the centroid is

$$\overline{x}=\frac{\int_0 ^b \frac{h}{b}x^2}{\int_0 ^b \frac{h}{b}x}$$
So the x-coordinate of the centroid is

$$\overline{x}=\frac{\int_0 ^b \frac{h}{b}x^2}{\int_0 ^b \frac{h}{b}x}$$

This is correct so far I assume, but what I do not understand is how to get the y-coordinate which should be the same answer.

EDIT: If my latex is wrong, I will type it over, so far the preview is only showing latex which I have typed for previous questions and not what I actually typed in the post, yet when I post the message it says my latex code is invalid.

 

[Dick]

I think everyone's latex is failing tonight. Must have to do with the server migration. Let's just try this again later.

 

[HallsofIvy]

The centroid of a triangle is simply the average of its three points. I searched through the code for your LaTex but I could find nowhere that you actually state what triangle you are talking about!