I can't believe it!!You have the limits reversed on the second one but other than that:
You have now successfully actually calculated a convolution integral. Now that you have done that you might enjoy an animated picture of a convolution, albeit with two different functions. But it is the same idea. The shaded area represents the value of the integral. Look here:
So can I just check I have understood it. So to find the integral I need to say how the limits change depending on whether x>0 or x<0 and then calculate the integral for each case?