There are 2 types of convolution:
The limit of the integral is from minus infinity to plus infinity
The limit is from zero to t.
When we use the first and when we use the second?
$$\int f(\tau)g(t-\tau)\; d\tau$$
the second kind of convolution in my question there is in page 346 of the book titled "elementary differential equations and boundary value problems" by William Boyce and Richard DiPrima. You can see the picture below.