Find the area bounded by $y = e^{x}$, $y = e^{2}$ and $x = 0$
I found the area by integrating the function $e^2 - e^x$ with the x-bounds of $0$ to $2$, which turned out to be $e^{2} + 1$.
$$\int_0^2 (e^2 - e^x)dx$$
However, the answer that is given in the packet is $e^{4} + e - 2e^{2}$. I can't figure out it the integral that I set up is wrong or if the answer that is given in the packet is wrong. Any help would be appreciated. Thanks in advance.
The answer provided by your book is wrong. As John Wayland Bales pointed out in the comments section, you should have gotten the following:
$$ \int_{0}^{2}\left(e^2-e^x\right)\,dx=\left[e^2x-e^x\right]_{0}^{2}= 2e^2-e^2-\left(0-e^0\right)=\\ 2e^2-e^2-(-1)=e^2+1. $$