The points (x,y,z) of this solid projected on the plane z=0 are in the parallelogram (2,1),(6,-1),(7,0),(3,2) and $\le z \le e^{x+2y}$.
$$\int_{2}^{7}dx \int_{-0.5x+2}^{-0.5x+3.5}dy \int_{0}^{e^{x+2y}}dz$$ is right?
the result of my book is $2(e^7-e^4)$
The black bordered region is what you are supposed to be integrating over. The red extensions are the additional areas your integral is including.
You need to set up different limitations on $y$ for $2 < x < 3$ than for $3 < x < 6$, and different limitations again for $6 < x < 7$. It is helpful to express it as the sum of 3 integrals, rather than a single integral.
An alternative method is to do a change of variables $(u, v) = g(x,y)$ for an appropriate mapping $g$ that will convert this parallelogram into a rectangle in $(u,v)$ space with horizontal and vertical edges. I'll leave it to you to figure out what expression of $u,v$ in terms of $(x,y)$ will do that.