A metal plate of constant density has a shape bounded by the curve y=sqrt(x), the x-axis, and the line x=1. Please help trying to study for exam tomorrow!

Question

ranga · Answer

The bounded area looks like this.  The enclosed area is the area of interest.
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megannicole51 · Answer

okay:D got that so far!

kmeis002 · Answer

Are you trying to find the weight?

ranga · Answer

There is no question in the problem.  Just a statement!

megannicole51 · Answer

Oh im sorry...im just trying to find the mass of the plate and the center of the mass

ranga · Answer

You need to integrate the function y = sqrt(x) between the limits x = 0 and x = 1 to get the area under the curve.  Assume the plate has a constant thickness t and density d then 
Volume = Area x t
Mass = Density x Volume = d x Area x t
Substitute Area from the integration.

megannicole51 · Answer

integration of sqrt(x)=(2/3)

megannicole51 · Answer

hey @ranga i just looked at the answer sheet and apparently they messed up and didnt add the density....so the constant density is 5

megannicole51 · Answer

attachment

megannicole51 · Answer

so the mass im good with:) its the center mass that i am confused with....i know that you have to either have x, y, or z when you are given the constant density, however, i only know how to do x

ranga · Answer

Oh, the density is given as mass per surface area and not as mass per volume.  So you can ignore the thickness t.

megannicole51 · Answer

yeah i realized that lol:)

ranga · Answer

There is a standard formula for finding the x and y coordinates of the center of mass which is what they are using.

megannicole51 · Answer

do u know what they are?

megannicole51 · Answer

and x im a little confused on how they are getting their answer...like i know how to set it up but solving it is where i get mixed up

ranga · Answer

$$X = \frac{ 1 }{ A } \int\limits_{a}^{b}xf(x)dx$$

X is x coordinate of center of mass and A is the area of the enclosed region found earlier.

ranga · Answer

$$Y = \frac{ 1 }{ A}\int\limits_{a}^{b}\frac{ 1 }{ 2}(f(x))^{2}dx$$

megannicole51 · Answer

so can you show out the steps to x and y for me so i can see it being done please?

ranga · Answer

$$A = \int\limits_{a}^{b}f(x)dx$$

megannicole51 · Answer

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so this is where i get stuck...