Calculating the are between 2 functions, by y-axis So, I have these 2 equations:
$$
y_1=e^{2x}\\y_2=-e^{2x}+4
$$
And I need to calculate the area they have limited right of the y-axis
this is how it looks: 
I have converted the formulas to x:
$$
x_1 = \frac{\log(y_1)}{2}\\x_2 = \frac{\log(4-y_2)}{2}\\
$$
Calculated the limits:
$$
\frac{\log(y_1)}{2}= \frac{\log(4-y_2)}{2}\\y = 2 \Rightarrow x = \frac{\log(2)}{2}
$$
And I got to the point of integrating:
$$
\int_0^{\frac{log(2)}{2}}(\frac{\log(4-y)}{2}- \frac{\log(y)}{2})dy = \\=\int_0^{\frac{log(2)}{2}}(\frac{\log(4-y)}{y})dy
$$
And I don't know how to integrate $\log(x)$ can you please help me?
 A: You don't need to inverse the function. Instead do the following:
Find the intersection by solving: $y_1=y_2$
Integrate $y_2-y_1$ from 0 to the intersection. This will give you the area.

Full calculation:
Solving $y_1=y_2$:
$$e^{2x}=-e^{2x}+4$$
$$2e^{2x}=4$$
$$e^{2x}=2$$
$$2x=\ln(2)$$
$$x=\frac12\ln(2)=\ln(\sqrt2)$$
$$y=e^{2x}=e^{2(\frac12\ln(2))}=2$$
Integrating the difference:
$$\int_{0}^{\frac12\ln(2)}(4-e^{2x}-e^{2x})dx=\int_0^{\frac12\ln(2)}4dx-\int_0^{\frac12\ln(2)}2e^{2x}\\
=2\ln(2)-[e^{\ln(2)}-e^{2*0}]=2\ln(2)-2+1=2\ln(2)-1$$
A: To bypass your problem (I dunno how to integrate $\log$ too $\ddot \smile$)
$$\begin{align}
\text{At point of intersection,$y_1=y_2$}&\implies x=\frac{\ln 2}{2}\\
\text{Area}&=\int_{0}^{\large\frac{\ln 2}{2}}\left[{(-e^{2x}+4)-e^{2x}}\right]\ \mathrm dx\\
&=\int_{0}^{\large\frac{\ln 2}{2}}\left({4-2e^{2x}}\right)\ \mathrm dx\\
\end{align}$$
A: If you're doing the integral over $y$ then your limits should be on $y$, which are from $1$ to $3$.
However, you don't need to change variables.  Note that the intersection occurs at $y = 2$, which is at $x = (1/2) \ln 2.$
Then your integral is
$$I = \int_0^{\frac{\ln 2}{2}} (-e^{2x} + 4)dx - \int_0^{\frac{\ln 2}{2}}e^{2x}dx.$$
This works out to $$\left.-\frac{e^{2x}}{2}+4x\right|_0^{\frac{\ln 2}2}-\left.\frac{e^{2x}}{2}\right|_0^{\frac{\ln 2}2} = -1 + 2 \ln 2.$$
A: When in doubt go to integral tables: 
http://en.wikipedia.org/wiki/List_of_integrals_of_logarithmic_functions
