Consider finding the center of mass of the substance residing in the region $$D=\{(x,y)|0\leq x\leq \ln 2,0\leq y \leq e^{x}\}$$ with density $\rho(x,y)=y$.

I write the total mass as

$$m=\int_{0}^{\ln 2}\int_{0}^{e^{x}}ydydx$$

and the $x$ coordinate as

$$x=\frac{\int_{0}^{\ln 2}\int_{0}^{e^{x}}xydydx}{m}$$

and the $y$ coordinate as

$$y=\frac{\int_{0}^{\ln 2}\int_{0}^{e^{x}}y^{2}dydx}{m}$$

The limits of the integrals most likely are wrong however since $e^{x}$ only goes from $1$ to $2$ when $x$ goes from $0$ to $\ln 2$.

How do I find the right limits?


The limits are ok as when you do the first integral involving the variable $x$, you will obtain a function of x, so when integrating through x you will obtain the range of values for $e^x$ correct. For example, for the mass:

$m=\int_0^{\ln2}\int_0^{e^x}ydydx= \int_0^{\ln2}[\frac{y^2}{2}]_0^{e^x}dx=\frac{1}{2}\int_0^{\ln2}e^{2x}dx=\frac{1}{2}[\frac{1}{2}e^{2x}]_0^{\ln2} =\frac{1}{4}(4-1)=\frac{3}{4}$

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    $\begingroup$ Shouldnt it be $m=3/4$ (it seems your last exponential should be $e^{2x}$ not $e^{x}$)? $\endgroup$ – user35202 Aug 17 '16 at 14:26
  • $\begingroup$ You are right I just edited $\endgroup$ – Josu Etxezarreta Martinez Aug 17 '16 at 14:32

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