How to solve -12+7x=x
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I'm not sure what is meant by "partitioning the <em>x</em>-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 <em>y</em> = 4<em>e</em> ˣ and <em>y</em> = -<em>x</em> + 4 intersect when <em>x</em> = 0 at the point (0, 4). Both curves then meet the vertical line <em>x</em> = 4. The exponential function is increasing while the linear one is decreasing, so 4<em>e</em> ˣ ≥ -<em>x</em> + 4.
The region is then the set of points,
<em>R</em> = {(<em>x</em>, <em>y</em>) | 0 ≤ <em>x</em> ≤ 4 and -<em>x</em> + 4 ≤ <em>y</em> ≤ 4<em>e</em> ˣ}
so the area is given by the integral,