1st Moment of Area, Double Integrals

simpsoncan2
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Hello I am in a statics course and am having difficulty solving a problem. In this problem I am supposed to find the 1st moment of the area. I have attached some photos of the work I have completed but it differs from the way my professor does these problems.

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His method for solving the double integrals is different from the method that I remeber and what my Cal book states as the proper way. Here is an example of how he is doing double integrals for the same problem, these are straight from my inclass notes:

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If someone has some insight on which method is correct and what I should do differently to solve my moment of an area problem I would really appreciate some help.

Thanks, Jacob
 
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Oh and the area/plate on the graph is homogeneous.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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