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Suppose I give you a curve

[tex]f(x) = \sin^2 (x) + ln(x)[/tex]

And suppose I tell you to rotate this curve about the x axis, we get disks. Now, I ask you, what is the center of mass of this object?

Now immediately, you could say that [tex]\bar{y} = 0[/tex] because it is symmetric about the x-axis. I don't argue, and I say you are right. But what about [tex]\bar{x}[/tex]?

It is actually NOT the same as the 2D- lamina. In fact

[tex]\bar{x} = \frac{\int_{a}^{b} \pi x[f(x)]^2 dx}{\int_{a}^{b} \pi [f(x)]^2 dx}[/tex]

Okay, so what am I trying get here?

Evaluate the integral of the curve from x = 1 to x = 10

[tex]\int_{1}^{10} \pi x (\sin^2 (x) + ln(x))^2 dx[/tex]

Do this (do it on a computer, trust me on this) and you get some nonreal numbers. I asked my TA about it and he said when I expand [tex]f(x) = \sin^2 (x) + ln(x)[/tex], I get a term that I won't know how touch until I get to analysis.

On Maple I got

[PLAIN]http://img143.imageshack.us/img143/9805/comh.jpg [Broken]

I asked my TA what it means physically and he said that he wasn't sure and maybe the complex/imaginary part meant it is significantly 0

So what does having an imaginary part mean?

[tex]f(x) = \sin^2 (x) + ln(x)[/tex]

And suppose I tell you to rotate this curve about the x axis, we get disks. Now, I ask you, what is the center of mass of this object?

Now immediately, you could say that [tex]\bar{y} = 0[/tex] because it is symmetric about the x-axis. I don't argue, and I say you are right. But what about [tex]\bar{x}[/tex]?

It is actually NOT the same as the 2D- lamina. In fact

[tex]\bar{x} = \frac{\int_{a}^{b} \pi x[f(x)]^2 dx}{\int_{a}^{b} \pi [f(x)]^2 dx}[/tex]

Okay, so what am I trying get here?

Evaluate the integral of the curve from x = 1 to x = 10

[tex]\int_{1}^{10} \pi x (\sin^2 (x) + ln(x))^2 dx[/tex]

Do this (do it on a computer, trust me on this) and you get some nonreal numbers. I asked my TA about it and he said when I expand [tex]f(x) = \sin^2 (x) + ln(x)[/tex], I get a term that I won't know how touch until I get to analysis.

On Maple I got

[PLAIN]http://img143.imageshack.us/img143/9805/comh.jpg [Broken]

I asked my TA what it means physically and he said that he wasn't sure and maybe the complex/imaginary part meant it is significantly 0

So what does having an imaginary part mean?

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