Find the centroid of the solid region ?

[CalleighMay]

Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

 

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

 

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

 

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

 

The problem is on pg 1033 in chapter 14.6 in the text, number 44. It reads:

 

Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Assume uniform density.

And it gives:

y=sqrt(4-x^2), z=y, and z=0

The question also asks to find the tripple integrals but he said that's WAY over our heads lol

 

and i looked at every other problem in this problem set and i don't understand a word of it. I looked at other worked out examples and they too make no sense to me I would attempt this one myself but i am literally stumped on this one 100%

 

Can anyone help me with this one? Thanks guys

[psychlone]

Hi CalleighMay,

 

I can solve you problem assuming that z= z(bar) = 0 (the bar goes above the z) which is the centroid or the centre of mass about z = 0. Then I can solve your problem.

 

y(bar) = Mxz/V --------- Where V = Volume and Mxz is the moment in the xz plane.

 

dMxz = y dV

 

dMxz = y Pi (r^2 - y^2) dy - Where r = the radius - the geometry of the equation a circle. Therefore 4 in your original equation is the radius squared which is 2.

 

Mxz = Int(dMxz) - Where int = integral.

 

the integral of dMxz = Pi r^4 /4 - WHere Pi = 3.141592654.......

 

V = 2/3 Pi r^3

 

y(bar) = Mxz/V = 3/8 r

 

Now you have to do the same for x(bar)

 

and build an equation for Myz in the yz plane.

 

Bearing in mind that the 3 dimensional shape your solving for is a hemishpere, with the y axis travelling up through the top of the dome.

 

Good luck.

Edited December 16, 2008 by psychlone
multiple post merged