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

[Jon1436]

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.

 

[Mark44]

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?

 

[Jon1436]

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

 

[Mark44]

OK, so you have this integral-
$$\int_{-r}^r \rho x \sqrt{r^2 - x^2}dx$$

A simple substitution can be used to evaluate this integral.

 

[Jon1436]

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?

 

[Mark44]

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.

 

[Jon1436]

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.

 

[Mark44]

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.

 

[Jon1436]

Alright I am going to try and do it from here thanks so much for your help.

 

[Mark44]

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.

 

[ideasrule]

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.