What is the Limit of the Centroid of an Area Bounded by the x-Axis and 1-x^n?

In summary, the conversation discusses finding the centroid of a bounded region and taking the limit as n approaches infinity. The formula for a centroid is given and the steps for finding the x- and y-coordinates are explained. The person asking for help is struggling with their calculations and is advised to show their work for further assistance.
  • #1
tinylights
18
0

Homework Statement


Okay, so the idea here is to take the centroid bounded by the x-axis and 1-x^n. N should be an even and positive integer. We should take the limit as n approaches infinity of both the x- and y-coords of the centroid, hopefully ending up with (0, 1/2).

Homework Equations


Formula for a centroid: A = integral of f(x), Mx = integral of xf(x), My = integral of f(x)^2 over 2. All of them should be definite integrals with the same "a" and "b" - I picked 0 and 1 and multiplied the whole thing by two since it appears to be an even function.

Then you divide Mx/A for your x-coord, My/A for your y-coord, and take the limit.

The Attempt at a Solution



See above. I am doing these steps over and over again but keep ending up with the same weird things. I get (n+1)/2(n+2) for my Mx/A, and this long convoluted thing for My/A - 1-2n+1/n+1 + 1/2n+1 / 2(1 - 1/n+1). This would be fine except the limit for Mx/A seems to be 1/2, when it really should be 0, and I'm certainly not getting a limit of 1/2 for My/A.

Thanks for your help guys.
 
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  • #2
Without seeing all your steps, its kind of hard to see where you could have gone wrong. It may be algebra. If you could; could you show how you got those expressions for Mx/A and My/A? Pictures of your work would suffice.
 

FAQ: What is the Limit of the Centroid of an Area Bounded by the x-Axis and 1-x^n?

What is the definition of "Limit of a centroid of area"?

The limit of a centroid of area refers to the point where the weighted average of an object's area is located. In other words, it is the point where the object would balance if it were placed on a fulcrum.

How is the limit of a centroid of area calculated?

The limit of a centroid of area is calculated by finding the average of the x and y coordinates of all the points within the object's boundary. This can be done using the formula: x̄ = ∫x * dA / ∫dA and ȳ = ∫y * dA / ∫dA, where x and y represent the coordinates of each point and dA represents the differential area.

What does the limit of a centroid of area tell us about an object?

The limit of a centroid of area provides information about the object's distribution of mass. It tells us where the object's center of mass is located, which can be useful in understanding its stability and how it will behave when subjected to external forces.

Can the limit of a centroid of area be outside of the object's boundary?

No, the limit of a centroid of area must always fall within the object's boundary. This is because it represents the center of mass of the object, which is inherently contained within its boundaries.

How is the limit of a centroid of area used in practical applications?

The limit of a centroid of area is used in various fields such as engineering, architecture, and physics to calculate the stability and balance of structures and objects. It is also used in designing and optimizing structures for maximum efficiency and safety.

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