Master Calculus with Expert Help: Finding the Centroid of f(x)=sqrt(r^2-x^2)

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In summary, the conversation revolves around finding the centroid of an equation in calculus. The person is seeking help because their teacher is not effective in teaching the subject. They are given a function and asked to find the centroid using a specific formula. The conversation also touches on the use of density and integration in the process. The expert advises using a substitution and recognizing the relationship between mass and the area under the curve. The conversation ends with the person thanking the expert for their help and planning to continue from that point.
  • #1
Jon1436
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Hello,

Here is my situational plea. My calculus teacher is horrible at teaching and god help me i have tried to get help in every position possible, so i am now hoping you guys can help me. I love calculus and I am sure i would i enjoy it more if i had a better teacher.

The problem is I have to find the Centroid of the equation f(x)=sqrt(r^2-x^2) between the intervals of -r and r.

Thankyou for your time in advance please give detailed steps if possible.
 
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  • #2
The centroid has two coordinates. Pretty obviously, the x-coordinate for your region is 0. What formula can you use to find the y-coordinate of the centroid?
 
  • #3
What my teacher told me was to use the mass total (M)x over mass to find the y coordinate.

so y=(Density)(x)f(x)dx/(Density)f(x)dx

Im not sure how to make the greek letter row on here so i just put density in parentheses
 
  • #4
OK, so you have this integral-
[tex]\int_{-r}^r \rho x \sqrt{r^2 - x^2}dx[/tex]

A simple substitution can be used to evaluate this integral.
 
  • #5
looks like it to me besides the fact that my teacher told me to put the integral you gave me over mass with mass being

(Density)f(x)dx

Do i even need to use the mass?
 
  • #6
Yes, you need the mass. The integral I wrote was the numerator. The denominator integral doesn't actually have to be done using calculus, as it represents rho times the area under the curve, which you can get if you know a very small amount of geometry.
 
  • #7
ok i have found the anti derivative of the numerator and now i must find the mass using the equation already mentioned. I am not given rao though I am only given the function f(x) so how do I go about finding the mass now. Sorry if I am asking dumb questions it just tells you how lost my teacher has led me to be.
 
  • #8
You'll have rho (not rao) in the numerator and denominator, so it cancels. Since rho is a constant, you can bring it out of both integrals.
 
  • #9
Alright I am going to try and do it from here thanks so much for your help.
 
  • #10
The integral you have for mass is harder than the one in the numerator, so you can make your life easier by recognizing that mass = rho * the area under the curve. I said it already, but it bears repeating.
 
  • #11
Draw the curve out and you'll see what the integral is. y=sqrt(r^2-x^2), so y^2=r^2-x^2.
 

FAQ: Master Calculus with Expert Help: Finding the Centroid of f(x)=sqrt(r^2-x^2)

What is the purpose of finding the centroid of a function?

The centroid of a function is the point at which the entire area under the curve is balanced. It is an important concept in calculus as it helps us understand the distribution of mass or density in a given area and can be applied in various real-world scenarios, such as determining the center of mass of an object.

How do I find the centroid of a function?

To find the centroid of a function, you will need to use the formula: x̄ = (1/A)∫x*f(x)dx, where A is the total area under the curve and f(x) is the function. This involves finding the integral of the function, which can be solved using various techniques in calculus such as integration by parts or substitution.

Can I use a calculator to find the centroid?

Yes, you can use a calculator to find the centroid, but you will still need to know how to set up the integral and understand the concept behind it. Calculators can help with the numerical calculation of the integral but they cannot solve it for you.

What is the importance of finding the centroid in real-world applications?

Finding the centroid can be applied in various real-world scenarios, such as determining the center of mass of an object, calculating moments of inertia in physics, and finding the average location of a distribution in statistics. It is also used in engineering and architecture to determine the balance and stability of structures.

Is finding the centroid a difficult concept to understand?

It can be challenging to understand at first, but with practice and a solid understanding of calculus concepts, it can become easier. It is important to fully understand the concept and the steps involved in finding the centroid to apply it correctly in different scenarios.

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