Originally Posted by
Jameson I think the region you're supposed to evaluate starts at y=1, has a right edge where the green and red graphs intersect, and an upper bound where the green graph crosses the y-axis again, which should be at y=7.
So because of the way the region is set up, we should integrate with respect to y instead of x. That means we need to solve both equations for x.
$\displaystyle (y=e^x) \rightarrow (x=\ln(y))$
$\displaystyle (y=7-e^x) \rightarrow (x=\ln(7-y))$
So looking at the region it looks like it's good to break it up into two smaller regions. From y=1 to where the graphs intersect, the x-values of the region are defined by $\displaystyle x=\ln(y)$ and after they intersect until y=7 the x-values are defined by $\displaystyle x=\ln(7-y)$
So let's figure out where they intersect.
$\displaystyle \ln(y)=\ln(7-y)$
$\displaystyle y=7-y$
$\displaystyle y=\frac{7}{2}$
So I think the whole are is:
$\displaystyle \int_{1}^{\frac{7}{2}} \ln(y)dy + \int_{\frac{7}{2}}^{7} \ln(7-y)dy$
Does that make sense? Hopefully I don't have any huge errors.