- #1

karush

Gold Member

MHB

- 3,269

- 5

\begin{align*}\displaystyle

I&=\int_0^1 \int_2^{2e^x}f(x,y) \quad dy \, dx

\end{align*}

$\textit{From the integral we have that}$

$$0\leq x\leq 1 \quad \textit{and} \quad 2\leq y\leq 2e^x$$

$\textit{So, we get that}$

$$y\leq 2e^x \Rightarrow \frac{y}{2}\leq e^x \Rightarrow \ln \left (\frac{y}{2}\right )\leq x$$

$\textit{Therefore, we get that}$

$$\ln \left (\frac{y}{2}\right )\leq x\leq 1 \quad

\textit{and} \quad 2\leq y\leq 2e^x\leq 2e^1=2e$$

$\textit{So, by changing the order of integrals we get the following}$

$$I=\int_2^{2e} \int_{\ln \left (\frac{y}{2}\right )}^1f(x,y) \quad dx \, dy $$

Ok just see if this is ok

presume this is as far as we can go.