Everything checks out to me, including your final answer.
I would be most grateful if someone could check my work.
Find the area enclosed by the graph of this function and the x axis between x=0 and x=6.
As the graph crosses the x-axis at x=0 and x=3 all values of x are positive between x=0 and x=3 all values of x>3 are negative.
It is necessary to to find the positive area bounded by x=0 and x=3 and subtract it from the negative area bounded by x=3 and x=6.
After hopefully performing the integration correctly I get an answer of 4.1319
Also, if you're allowed to do these problems on the calculator I would suggest using the absolute value button. It's for you to note that total area means positive and negative area, not just the integral, but if you need a quick calculation, I know on the TI models you can enter