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 ;)

 

[Defennder]

Check out the double integral formulae for center of mass. It goes like this:
$$\bar{x} = \frac{M_y}{M}$$

$$\bar{y} = \frac{M_x}{M}$$

where $$M = \iint_R \delta (x,y) dA$$.

where $$\bar{x}, \bar{y}$$ refers to the coordinates of the centre of mass.

 

[dynamicsolo]

I'm assuming that your calculus sequence is much like that where I work, so your Calculus I and II only deal with single-variable calculus. (I surmise this since your problem is supposed to be a "preview" of something in Calculus III.)

Multiple integration is the "big hammer" by which this problem can be easily beaten down. But when you don't have a particular tool available, you sometimes have to be clever instead.

Have you drawn a picture of this figure? I won't say just now what it looks like, but you should notice that it is symmetric about the y-axis. So you already know one coordinate of the centroid!

It remains to find the other two coordinates, using only single-variable integration. So we'll have to recast this problem accordingly. Look at the infinitesimal "slices" of this figure parallel to the yz-plane. What shape are they? (They will also all be similar, in the geometical sense of the word.) What do you know about the centroid of that shape? (If you haven't worked it out already, it will take only a minute to do so...)

We are going to integrate "slices" along the x-axis. Because of the figure's symmetry, we only need to integrate from x = 0 to x = (what?). [The other half makes a mirror-image contribution, which is unnecessary to evaluate for finding the centroid.] The "slices" have to be "weighted" with an infinitesimal mass dm, which is given by the density times the area of each slice (as a function of x) times dx.

So the coordinates of the centroid will be

$$x_C = 0$$

$$y_C = \frac{\int_{0}^{a} \rho \cdot y(x) \cdot A(x) \ dx }{M}$$

and

$$z_C = \frac{\int_{0}^{a} \rho \cdot z(x) \cdot A(x) \ dx }{M}$$

with
$$M = \int_{0}^{a} \rho \cdot A(x) \ dx$$ .

That's as much as I'm saying for now. You should find the upper limit a and work out the details before we can discuss this further on the Forum...

EDIT: I thought of an even more direct way to do this. Looking at the figure, think about the curve that the centroids of the "vertical slices" (parallel to the yz-plane) would sweep out in space. What shape is that? How do you find the centroid of such a curve? How would it be positioned in three dimensions? (Again, the problem can be worked out with single-variable calculus, and you may have already found the centroid of this curve in your earlier examples in lectures or the book or in homework problems...)