Graphical CT Convolution
Doing this graphically, or conceptually, is quite easy.  This problem seems to have been designed to make it easy.

Imagine the top signal sliding along, with the bottom signal staying fixed.  For each position of the top signal, produce the product of the two signals, then add up the resulting area under the curve.

For example, at T = 0, the top signal is as shown.  Note that only the part of the top signal between t=1 and t=2 matters.  The bottom signal is 1 there, so basically the product of the two is always the small window of the top signal from t=1 to t=2.

At T = 0, that product is just -1 from t = 1 to 2, for a total area of -1.  You should be able to see from inspection that with the top signal advanced by 1 to the right (T = 1), that same product will be a positive rectangle with area +1.  The values in between will obviously be linear from -1 to +1.  Likewise, at T = 2 you also get +1, then at T = 3 and greater, 0.

So basically you solve the convolution by inspection at three points, then also see from inspection that it will vary linearly between these point.