I'm not sure what is meant by "partitioning the x-axis" - partition is a term more often used with computing Riemann sums. So either partitioning has a special meaning for you, or you have to approximate the area with a Riemann sum.
I'm going to assume you just want the exact area of the given region. Notice that the two curves y = 4e ˣ and y = -x + 4 intersect when x = 0 at the point (0, 4). Both curves then meet the vertical line x = 4. The exponential function is increasing while the linear one is decreasing, so 4e ˣ ≥ -x + 4.
The region is then the set of points,
R = {(x, y) | 0 ≤ x ≤ 4 and -x + 4 ≤ y ≤ 4e ˣ}
so the area is given by the integral,
[tex]\displaystyle\int_0^4 (4e^x-(-x+4))\,\mathrm dx=\int_0^4(4e^x+x-4)\,\mathrm dx=\left(4e^x+\frac{x^2}2-4x\right)\bigg|_0^4=\boxed{4e^4-12}[/tex]
