A triangle with vertices (0,0), (0,1) and (1,2), counterclockwise, delimits a region. Determine an integral over this region of the following expression:
$$\int_{\Omega}(x-y)dx+e^{x+y}dy$$
This kind of exercises fit the line integral, but I don't know how to start solving it, I thought I'd use Green's Theorem. The key is to integrate x first, then y.
$$\int_{\Omega}(x-y)dx+e^{x+y}dy=\int_{\Omega}\left[\dfrac{\partial}{\partial x}(e^{x+y}) – \dfrac{\partial}{\partial x}(x-y)\right]\,dx\,dy =\int_{\Omega}\left[e^{x+y} +1\right]\,dx\,dy $$
What would the extremes of integration look like? I don't know how to continue
Best Answer
By green’s theorem, we can convert our line integral (which would require 3 integrals to solve) into a single double integral.
So far, your setup is correct, where we have $$\int_{\Omega}\left[e^{x+y} +1\right]\,dx\,dy$$ over the region(in green)
We can set up our double integral bounds then to be $$\int_0^1 \int_{2x}^{x+1} e^{x+y}+1\,dydx$$
Now all that is left is to evaluate the double integral, which should be pretty easy.
to set up bounds, we'll use the arrow and shadow method. Choose the $y$ axis first and draw arrows from $y=-\infty$ to $y=\infty$, and see what function the arrow passes through and exits from.
It passes through the yellow($2x$) first, and then exits purple($x+1$). Those are our y bounds. Now shine a light from $y=\pm\infty$ and see what shadow the region casts on the $x$ axis. This is just the interval $[0, 1]$. Those are our $x$ bounds.