Find the mass and center of mass of lamina ?

[CalleighMay]

Find the mass and center of mass of "lamina" ?

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 1015 in chapter 14.4 in the text, number 14. It reads:

Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density or densities (Hint: some of the integrals are similar in polar coordinates).
And it gives:
y=x^3, y=0, x=2, p=kx

(the p looks a little different, most likely represents something else, and the k really is a k, not to be confused with that wavelength symbol lol)

I looked at similar problems in the same section and came up with the following for this one:

I first graphed the equations on the same coordinates.
I think I am supposed to take the integral from (0 to 2) of the integral from (0 to x^3) kxdxdy? or kxdydx? (something about horix simple or vert simple??)

And then I'm confused as to what to do next, even if i was sure what i have so far is correct.

Any further help would be greatly appreciated. Thanks guys

 

[cristo]

Well, the first thing I would do is to sketch a graph so that you can tell whether you've got the correct limits of integration or not. Anyway, your integral $$\int_0^2 \int_0^{x^3}kxdydx$$ is correct, so now you need to evaluate it.

A few notes: firstly, the "p" you quote is actually the greek letter rho $\rho$, which stands for density. Secondly, the order of the differentials (i.e. dx and dy) inside the integral matters, since it determines which of the integration signs refers to integration with respect to x or with respect to y. You work from the inside out, so in the above case, the dy corresponds to the rightmost integration sign, etc.

 

[CalleighMay]

is that for Mx? Do i need My too? I just don't even know what to do to integrate that, what will that be finding and what would i do next?

I don't know how to integrate with the k and x in there...