Hello, rawkstar!
Since you didn't show your work, we have no idea where your error is.
$\displaystyle \text{Consider the region bounded by the graph of:}$
$\displaystyle y\:=\:x^3 + 18x^2 + 108x,\;(0 \le x \le 1)\,\text{ on the left,}$
$\displaystyle x\text{axis below, }\,y = 127\text{ above, and }x = 3\text{ on the right.}$
$\displaystyle \text{(1) Find the area of the region.}$
The graph looks like this:
Code:

127+ *        *
 .:::::::::::::::
 .*:::::::::::::::
 .*B:::::::A::::::::
 .*:::::::::::::::::::
  *    +    +    + 
 1 2 3

Area $\displaystyle \,A$ is a 2by127 rectangle: .$\displaystyle A = 254$
Area $\displaystyle \,B$ requires an integral: .$\displaystyle \displaystyle B \;=\;\int^1_0(x^3 + 18x^2 + 108x)\,dx$
We have: .$\displaystyle B \;=\;\frac{1}{4}x^4 + 6x^3 + 54x^2\,\bigg]^1_0 \;=\;\frac{1}{4} + 6 + 54$
. . . . . . . .$\displaystyle B \;=\;60\frac{1}{4}$
Therefore, the total area is: .$\displaystyle 254 + 60\frac{1}{4} \;=\;314\frac{1}{4}\text{ units}^2.$