Hi, can anyone please help me understand this distribution?

If X and Y are both uniformly distributed on (0,1), calculate the probabily of X+Y

Here's what I have:

fx(a)=fy(a)=1 because both uniformaly distributed.

Then fx+y(a)=integral (0,1) of fx(a-y)dy

just found by rearranging x+y=a and fy(a)=1

Now, it says that fx+y(a)=integral (0,a) dy=a

My question is how they changed the integration limits

Why isn't it fx+y(a)=integral (0,1) dy=1?

Intuitively the first answer makes more sense, but I'm just confused with the math.