Finding centroids when integrating with respect to y

In summary, the conversation discusses the topic of centroids of constant density regions in AP Calculus. The formulas for finding the x and y coordinates of the center of mass were mentioned, with a focus on integrating with respect to x. The question of what the formulas would be when integrating with respect to y was raised, but no specific example was given.
  • #1
pakmingki
93
1

Homework Statement


no specific problems

Homework Equations


x center of mass = moment about y/total area
y center of mass = moment about x/total area

The Attempt at a Solution


ok, so I am in AP calculus, and since the AP testing has ended, we've done some random topics, such as centroids of constant density regions. So far, we've only found centroids by integrating with respect to x.
So, the formula for the x center of mass coordinate is the [integral from a to b of x(f(x) - g(x))]/total area, and the y center of mass coordinate is [integral from a to b of 1/2 * (f(x)^2 - g(x)^2)]/total area.

Ok, but those formulas only apply to when you integrate with respect to x. What would the formulas be when you integrate with respect to y?

thanks.
 
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  • #2
I am not quite sure if I understand what you are asking. Could you give an example?
 

1. What is the definition of a centroid?

A centroid is the geometric center or average position of all the points in a shape or object. It is often referred to as the "center of mass" or "center of gravity" of the object.

2. How do you find the centroid of a shape when integrating with respect to y?

To find the centroid of a shape when integrating with respect to y, you first need to determine the limits of integration for y. These limits will be the upper and lower bounds of the shape. Then, use the formula for the y-coordinate of the centroid:
ȳ = (1/A) * ∫y * f(y) dy
Where A is the area of the shape and f(y) is the function that represents the shape's boundary. Integrate this formula to find the y-coordinate of the centroid.

3. What is the formula for finding the x-coordinate of the centroid?

The formula for finding the x-coordinate of the centroid is:
x̄ = (1/A) * ∫x * f(y) dy
This formula is similar to the one for the y-coordinate, but instead of integrating with respect to y, you integrate with respect to x. The limits of integration for x will also be the upper and lower bounds of the shape.

4. Can the centroid of a shape lie outside of the shape?

Yes, the centroid of a shape can lie outside of the shape. This can happen if the shape is not symmetrical or if the distribution of the shape's area is uneven. In such cases, the centroid will still be the center of mass of the shape, but it may not be within the bounds of the shape itself.

5. How is finding the centroid useful in real-world applications?

Finding the centroid is useful in a variety of real-world applications, including engineering, architecture, and physics. It can help determine the balance and stability of structures, calculate the center of gravity for objects, and aid in designing efficient and effective structures. It is also used in statistics to find the average or central tendency of a set of data points.

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